Lucas B Guillaume

Jérôme

Sonia

Antonio David

Florian, Sergi, William Alexandra Bastien

Keywords: Optimisation Stochastique, Descente de Gradient, Equation de Langevin, Dynamique de Langevin, Recuit Simulé, Apprentissage Machine, Apprentissage Profond, Réseaux de Neurones, Schéma d'Euler-Maruyama, Mesures de Gibbs, Contrôle Stochastique, Méthodes de Monte Carlo, Méthodes MCMC Stochastic Optimization, Gradient Descent, Langevin Equation, Langevin Dynamics, Simulated Annealing, Machine Learning, Deep Learning, Neural Networks, Euler-Maruyama Scheme, Gibbs Measures, Bayesian Inference, Stochastic Optimal Control, Monte Carlo Methods, MCMC Methods. 1.1. Stochastic algorithms Stochastic Optimization, Langevin Equation, Simulated Annealing, Neural Networks Stochastic Optimization, Langevin Equation, Simulated Annealing, Neural Networks. Chapter 4

Durant ma thèse, j'ai pu bénéficier de l'accueil chaleureux et du soutien que m'ont témoigné les membres du Laboratoire de Probabilités, Statistique et Modélisation (LPSM) de Sorbonne Université. Je remercie particulièrement Vincent Lemaire et Idris Kharroubi pour nos discussions mathématiques et pour avoir répondu à mes questions sur leurs articles, et Daphné Giorgi pour organiser le séminaire Infomath et pour son aide précieuse en informatique. Je remercie Nicolas Fournier et encore une fois Idris Kharroubi pour leur participation à mon comité de thèse en début de troisième année. Je remercie Olivier Pironneau, professeur au laboratoire voisin Jacques-Louis Lion, pour nos échanges sur un sujet commun qui m'ont été très utiles pour la rédaction d'un article s'inspirant d'un de ses travaux.

Je remercie Fabien Panloup, professeur à l'Université d'Angers, pour avoir accepté de coécrire un article avec Gilles Pagès et moi-même. Sa collaboration a été très précieuse pour établir un résultat important pour ma thèse.

Je remercie Arturo Kohatsu-Higa, professeur à

Bibliography xv

Notations

We give the notations that are commonly used throughout this thesis.

• We endow the space R d with the canonical Euclidean norm denoted by | • | and we denote ⟨•, •⟩ the associated canonical inner product. For x ∈ R d and for R > 0, we denote B(x, R) = {y ∈ R d : |y -x| ≤ R}.

• For x, y ∈ R d we denote (x, y) = {ux + (1 -u)y, u ∈ [0, 1]} the geometric segment between x and y.

• For a, b ∈ N we denote M a,b (R) the set of a × b matrices with real coefficients. If a = b we sometimes use the notation M a (R) = M a,a (R).

• For u, v ∈ R d we denote by u * v, or sometimes by u ⊙ v, the element-wise product (Schur product) i.e. u * v = [u i v i ] 1≤i≤d . Similarly we define u ⊘ v = [u i /v i ] 1≤i≤d .

• For u ∈ R d 1 and v ∈ R d 2 we denote by u⊗v the tensor product with a⊗b = [a i b j ] 1≤i≤d 1 ,1≤j≤d 2 .

• For d 1 , . . . , d r ∈ N we define R d 1 ⊗ . . . ⊗ R dr the set of tensors of order r with dimensions d 1 , . . . , d r i.e. tensors of the form (x i 1 ,...,ir ) 1≤i 1 ≤d 1 ,...,1≤ir≤dr .

• For M ∈ (R d ) ⊗k , we denote by ∥M ∥ its operator norm, i.e. ∥M ∥ = sup u∈R d×k , |u|=1 M • u.

If M : R d → (R d ) ⊗k , we denote ∥M ∥ ∞ = sup x∈R d ∥M (x)∥.

• For u ∈ R d we denote diag(u) ∈ M d (R) the diagonal matrix with diagonal u.

• For (A, d A ) and (B, d B ) two metric spaces, typically A = R d 1 and B = R d 2 endowed with the Euclidean distance, we denote C k (A, B) the set of functions f : A → B being k times differentiable with continuous k th derivative. If there is no ambiguity, we simply write f ∈ C k .

• For (A, d A ) and (B, d B ) two metric spaces, we say that f : A → B is coercive if d B (f (x), 0) → ∞ as d A (x, 0) → ∞ uniformly. For normed spaces, we can rewrite |f (x)| → ∞ as |x| → ∞.

• We denote C k b the set of functions in C k being bounded with bounded partial derivatives up to order k; we denote C k b the set of functions in C k with bounded partial derivatives up to order k but that are not necessarily bounded themselves.

• If f : R d 1 → R d 2 is Lipschitz continuous, we denote by [f ] Lip its Lipschitz constant.

• For f : R d 1 → R d 2 being C k , for 1 ≤ j ≤ k we denote by ∇ j f : R d 1 → R d 2 ⊗ (R d 1 ) ⊗j its derivative tensor of order j given by ∇ j f = ∂ j f i 0 ∂x i 1 . . . ∂x i j 1≤i 0 ≤d 2 , 1≤i 1 ,...,i j ≤d 1 .

• For f : R d → R such that min R d f exists, we denote

argmin(f ) = x ∈ R d : f (x) = min R d f .
• For f, g : R q → R and x 0 in R, we write f (x) ∼ g(x) as x → x 0 meaning f (x) = g(x) + o(g(x)) as x → x 0 .

• If u n and v n are two real-valued sequences, we write u n ≍ v n meaning that u n = O(v n ) and v n = O(u n ).

• For a random vector X, we denote by [X] its law.

• We denote the total variation distance between two probability distributions π 1 and π 2 on

R d : d TV (π 1 , π 2 ) = 2 sup A∈Bor(R d ) |π 1 (A) -π 2 (A)|.
We have as well

d TV (π 1 , π 2 ) = sup R d f dπ 1 - R d f dπ 1 : f : R d → [-1, 1] measurable .
Moreover, we recall that if π 1 and π 2 admit densities with respect to some measure reference λ, then

d TV (π 1 , π 2 ) = R d dπ 1 dλ - dπ 2 dλ dλ.
• We denote the L p -Wasserstein distance between two distributions π 1 and π 2 on R d :

W p (π 1 , π 2 ) = inf R d
|x -y| p π(dx, dy)

1/p : π ∈ P(π 1 , π 2 ) ,

where P(π 1 , π 2 ) stands for the set of probability distributions on (R d ×R d , Bor(R d ) ⊗2 ) with respective marginal laws π 1 and π 2 . For p = 1, let us recall the Kantorovich-Rubinstein representation of the Wasserstein distance of order 1 [Vil09, Equation (6.3)]:

W 1 (π 1 , π 2 ) = sup R d f (x)(π 1 -π 2 )(dx) : f : R d → R, [f ] Lip = 1 . (0.0.1)
• For some distance D on the set of probability distributions on R d and for X and Y two R d -valued random vectors, we denote without ambiguity D(X, Y ) = D([X], [Y ]).

• For x ∈ R d , we denote by δ x the Dirac mass at x.

• We generally consider (Ω, F, P, (F t ) t≥0 ) a filtered probability space satisfying the usual conditions.

• We denote by ReLU or (•) + the ReLU function: ReLU(x) = (x) + = max(0, x).

• For u, v ∈ R, we define u mod(v) = u -v⌊u/v⌋.

• We often use the notation C and c to denote positive constants, which may change from line to line.

Chapter 1

Introduction

Stochastic algorithms are a powerful tool to solve complex optimization and inference problems, where the state space is not explored deterministically but randomly. Stochastic algorithms have recently known a renewed interest, especially from the Machine Learning community; new research aims at reworking, modifying the existing stochastic algorithms in order to make them more efficient and to accelerate their convergence speed.

Stochastic algorithms 1.Stochastic gradient descent and the Robbins-Siegmund lemma

Let V : R d → R and let us consider the following optimization problem with no constraint:

Minimize

x∈R d V (x).
(1.1.1)

We assume that the dimension d is large so that this optimization problem is complex to solve. We generally make the following assumptions on the function V : V is C 1 , coercive i.e. V (x) → +∞ as |x| → +∞, which implies in particular that min(V ) exists. The stochastic gradient descent (SGD) algorithm is a gradient descent with noise:

X 0 ∈ R d , X n+1 = X n -γ n+1 (∇V (X n ) + ζ n+1 ) , (1.1.2)
where (γ n ) is a positive sequence of steps which can be constant or decreasing to 0 and (ζ n ) is a sequence of noise corresponding to noisy observations of the true gradient ∇V (X n ). Illustrations of the gradient descent are given in Figures 1.1 and 1.2; we hope that for large enough n, X n will be close to argmin(V ). A classical setting is where ∇V (X n ) is measured with no bias i.e. there exist a random variable Z with values in R q and a function v :

R d × R q → R such that ∀x ∈ R d , E[v(x, Z)] = V (x) and E[∂ x v(x, Z)] = ∇V (x).
(1.1.3)

The stochastic gradient algorithm then becomes:

X n+1 = X n -γ n+1 ∂ x v(x n , Z n+1 ), (1.1.4)
where (Z n ) is i.i.d. and where Z 1 ∼ Z. The noise sequence ζ is then a sequence of increments of a martingale i.e. for all n ∈ N, E[ζ n+1 |X 0 , . . . , X n ] = 0. Stochastic algorithms were introduced by Robbins and Monro in [START_REF] Robbins | A stochastic approximation method[END_REF] in 1951. The Robbins-Siegmund Lemma [START_REF] Robbins | A convergence theorem for non negative almost supermartingales and some applications[END_REF] guarantees the convergence of the stochastic gradient algorithm under some conditions. Lemma 1.1.1 (Robbins-Siegmund Lemma). Let h : R d → R d and let H :

R d × R q → R d be such that ∀x ∈ R d , E[H(x, Z)] = h(x).
Assume that there exists a C 1 function V : R d → R + such that ∇V is Lipschitz-continuous and |∇V | 2 ≤ C(1 + V ) for some constant C > 0. Assume furthermore that ⟨∇V, h⟩ ≥ 0 and that

∀x ∈ R d , ∥H(x, Z)∥ 2 2 ≤ C(1 + V (x)).
Let (γ n ) be a sequence of positive real numbers such that

∞ n=1 γ n = +∞, ∞ n=1 γ 2 n < +∞.
Then the sequence (X n ) defined by

X n+1 = X n -γ n+1 H(X n , Z n+1 ) satisfies:
1. X n -X n-1 → 0 almost surely and in L 2 as n → ∞.

2. The sequence E[V (X n )] n∈N is bounded.

3. V (X n ) converges almost surely.

4. n≥1 γ n ⟨∇V, h⟩(X n-1 ) < +∞ almost surely.

5. The martingale sequence ( n k=1 γ k (H(X k-1 , Z k ) -h(X k-1 )) converges almost surely and in L 2 (P).

Classic examples of stochastic optimization problems

We now give classic examples where such optimization problems arise. Figure 1.2: Example of gradient descent in R 2 with different gradient descent algorithms for the function V (x, y) = -sin(x 2 ) cos(3y 2 )e -x 2 y 2 -e -(x+y) 2 with two global minima next to (0, 0). For each gradient descent method we indicate its learning rate in the legend.

Machine Learning and regression

Let us consider data samples u i ∈ R d in and y i ∈ R dout for 1 ≤ i ≤ M associated to a regression problem, where (u i ) are the inputs and (y i ) are the outputs. That is, we look for a function ψ : R d in → R dout which fits to the data i.e.

∀ 1 ≤ i ≤ M, ψ(u i ) ≈ y i .
In other words, the objective is to extract a model from the empirical data. We look for a function ψ in a family of functions parametrized by a finite-dimension parameter: {ψ x , x ∈ R d }. For a loss function L : R dout → R + which measures the error between the prediction ψ x (u i ) and the true data y i , the regression problem then becomes the minimization of the average loss over the dataset and can be written as the following optimization problem:

Minimize x∈R d V (x) := 1 M M i=1
L(ψ x (u i ) -y i ).

(1.1.5)

Examples of parametrization:

• Least-square regression: Let us write x = (θ, ι) ∈ M dout,d in (R) × R dout so that ψ x is the affine function ψ (θ,ι) (u) = θ • u + ι and L is the quadratic loss i.e. L(w) = (1/2)|w| 2 .

• Logistic regression: Let us assume that y i ∈ {±1} are labels associated to a binary classification problem and that u i ∈ R d . Then the problems becomes Minimize

x∈R d V (x) := 1 M M i=1
log 1 + e -y i ⟨x,u i ⟩ .

• Fully connected artificial Neural Networks: Let φ : R → R be a non-linear function. The function φ is generally chosen to be a sigmoid-type or a ReLU-type function, see Figure 1.4. Let K + 1, K ∈ N, be the number of layers of the neural network and for k = 0, . . . , K let d k ∈ N be the size of the k th layer with d 0 = d in and d K = d out . For k = 1, . . . , K, u ∈ R d k-1 and for θ k ∈ M d k ,d k-1 (R) and ι k ∈ R d k we define the vector in R d k :

φ θ k ,ι k (u) := φ([θ k • u + ι k ] i ) 1≤i≤d k ,
i.e. the scalar function φ is applied to the vector θ k • u + ι k coordinate by coordinate. Let us write x = (θ 1 , ι 1 , . . . , θ K , ι K ), then the output of the neural network is

ψ x (u) = θ K • φ θ K-1 ,ι K-1 • • • • • φ θ 1 ,ι 1 (u) + ι K .
An illustration of a fully connected neural network is given in Figure 1.3. Neural networks are known for their ability to approximate a wide range of non-linear functions in high dimension and to fit to many non-linear regression tasks [START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF][START_REF] Lecun | Deep learning[END_REF].

The gradient descent algorithm for (1.1.5) then becomes

X n+1 = X n - γ n+1 M M i=1 ∂ x ψ x (u i ) • ∇L(ψ x (u i ) -y i ).
However, if the number of samples M is large then computing ∇V exactly at each iteration is costly. We instead replace this algorithm by a stochastic gradient algorithm: at each iteration n let i n ∈ {1, . . . , M } be a random index chosen uniformly at random and the iteration becomes As a more stable version of (1.1.6) and still computationally efficient, the gradient can be estimated by an average over a small batch of data:

X n+1 = X n - γ n+1 M ∂ x ψ x (u i n+1 ) • ∇L(ψ x (u i n+1 ) -y i n+1
X n+1 = X n - γ n+1 N batch i∈I n+1 ∂ x ψ x (u i ) • ∇L(ψ x (u i ) -y i )
where I n+1 is a subset of {1, . . . , M } of size N batch taken uniformly at random with N batch ≪ M . An example of neural network training is given in Figure 1.5.

Computation of quantiles and Value at Risk

Let Z be a random variable taking its values in R and such that E|Z| < ∞. For α ∈ [0, 1], let q α be the quantile of order α, i.e.

q α = inf{u ∈ R : P(Z ≤ u) ≥ α}.

In the Mathematical Finance community, the quantile q α is called Value at Risk (VaR) and is widely used with α close to 1 as a risk measure for risk management [START_REF] Jorion | Value at Risk: A New Benchmark for Measuring Derivative Risk[END_REF]. Following [START_REF] Uryasev | Conditional value-at-risk: optimization approach[END_REF] we have the characterization

q α = argmin x∈R E x + 1 1 -α (Z -x) + =: argmin x∈R E[v(x, Z)].
To compute q α , one may apply a gradient descent algorithm however computing exactly the expectation may be impossible. Instead the stochastic gradient algorithm reads

X n+1 = X n -γ n+1 1 if X n ≥ Z n+1 -α/(1 -α) if X n < Z n+1 ,
where (Z n ) is an i.i.d. sequence and where Z 1 ∼ Z. The Conditional Value at Risk (CVaR) is then defined as

E Z|Z ≥ q α = E[v(q α , Z)]
and can be directly estimated by the online computation [START_REF] Bardou | Computing VaR and CVaR using stochastic approximation and adaptive unconstrained importance sampling[END_REF]:

1 Γ n n k=1 γ k v(X k , Z k ) with Γ n := γ 1 + . . . + γ n .

Stochastic optimal control

Let us consider the following stochastic control problem associated to an SDE in continuous time:

min u J(u) := E T 0 G(t, Y u t )dt + F (Y u T ) , (1.1.7) dY u t = b(Y u t , u t )dt + σ(Y u t , u t )dW t , t ∈ [0, T ] (1.1.8)
where b : R

d 1 × R d 3 → R d 1 , σ : R d 1 × R d 3 → M d 1 ,d 2 (R), W is a R d 2 -valued Brownian motion
and u is a R d 3 -valued continuous adapted process, T > 0, G : R + × R d 1 → R and F : R d 1 → R. Stochastic control problems are usually solved using specific strategies, such as Forward-Backward SDEs (FBSDEs) [START_REF] Peng | Fully coupled forward-backward stochastic differential equations and applications to optimal control[END_REF], or by solving Hamilton-Jacobi-Bellman (HJB) optimality conditions [START_REF] Bellman | Dynamic programming[END_REF] through Partial Differential Equation (PDE) methods or by stochastic dynamic programming [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF]. Such problems can also be solved using Neural Networks calibrated 1.1. Stochastic algorithms by SGD techniques [GM05, HE16, WLP + 19, CL21, BHLP22]. More specifically, we proceed as follows. We approximate the control u as the output of a (fully connected) neural network

u t = u θ (t, Y u t ) (1.1.9)
where u θ is a neural function with finite-dimensional parameter θ ∈ R d . Since (1.1.8) defines a Markovian process, we can assume that u t depends only on t and on Y u t instead of t and the whole previous trajectory (Y u s ) s∈ [0,t] . We consider a subdivision of [0, T ]:

0 = t 0 < t 1 < • • • < t N = T (1.1.10)
and approximate (1.1.7) by the time discretized model min

θ∈R d J(u θ ) := N -1 k=0 (t k+1 -t k )G(t k+1 , Ȳ u θ t k+1 ) + F ( Ȳ u θ t N ), (1.1.11) Ȳ u θ t k+1 = Ȳ u θ t k + (t k+1 -t k )b( Ȳ u θ t k , u θ (t k , Ȳ u θ t k )) + t k+1 -t k σ( Ȳ u θ t k , u θ (t k , Ȳ u θ t k ))ξ k+1 , (1.1.12) ξ k ∼ i.i.d.
N (0, I d 2 ).

(1.1.13)

For every θ ∈ R d , ∇ θ J can be computed by automatic differentiation, where the gradient w.r.t. to θ is tracked all along the trajectories through the recursive dynamics of the Euler scheme (1.1.12) [START_REF] Giles | Smoking adjoints: fast evaluation of Greeks in Monte Carlo calculations[END_REF][START_REF] Michael | Monte Carlo evaluation of sensitivities in computational finance[END_REF]. Then the SGD algorithms reads

θ n+1 = θ n -γ n+1 ∇ θ J(u θn , (ξ n+1 k ) 1≤k≤N ) (1.1.14)
where (ξ n k ) 1≤k≤N,n∈N is an array of i.i.d. random vectors N (0, I d 2 )-distributed, (γ n ) n∈N is a non-increasing positive step sequence and where we wrote the dependence of J in (ξ n k ).

Variance reduction in Monte Carlo simulation

In Monte Carlo simulation, we aim to estimate E[φ(Z)] where Z is a random vector taking its values in R d , φ : R d → R and such that E|φ(Z)| 2 < ∞, using the estimator 1 M M i=1 φ(Z i ) where M ∈ N is large and Z i are i.i.d. with Z 1 ∼ Z. Variation reduction techniques help to estimate E[φ(Z)] with smaller variance, yielding tighter confidence intervals.

More specifically, let us consider variance reduction by unconstrained recursive importance sampling as introduced in [START_REF] Arouna | Adaptative Monte Carlo method, a variance reduction technique[END_REF] and [START_REF] Lemaire | Unconstrained recursive importance sampling[END_REF]. Let us assume that Z ∼ N (0, I d ). For example, for a Call option on a basket of risky assets driven by a multi-dimensional Black-Scholes model we have

φ(Z) =   d i=1 a i s i 0 exp     r - d j=1 σ 2 ij 2   T + √ T d j=1 σ ij Z j   -K   + ,
where r, T , K ∈ R + , σ ∈ M d (R) is definite positive definite symmetric and s 0 , a ∈ (0, s 0 ) d and Z = (Z 1 , . . . , Z d ).

For θ ∈ R d we have V (θ) := E φ 2 (Z + θ)e -|θ| 2 -2⟨θ,Z⟩ = E φ 2 (Z)e -⟨θ,Z⟩+|θ| 2 /2 so that to estimate E[φ(Z)] by Monte Carlo simulation using (1.1.15) with a smaller variance. The (naive) stochastic gradient algorithm writes θ n+1 = θ n -γ n+1 φ 2 (Z n+1 )e -⟨θn,Z n+1 ⟩+|θn| 2 /2 (θ n -Z n+1 ).

E[φ(Z)] = 1 (2π) d/2 R d φ(z)e -|z| 2 2 dz = 1 (2π) d/2 R d φ(z + θ)e
(1.1.16)

However (1.1.16) suffers from instability and explosion because of the factors of exponential growth. This algorithm is improved in [START_REF] Lemaire | Unconstrained recursive importance sampling[END_REF] using the following new representation of the gradient:

∇V (θ) = E φ 2 (Z -θ)(2θ -Z) .

Optimal quantization

Vector quantization consists in mapping input values from a large (or continuous) set to output values in a (finite) smaller set. It and was originally introduced for data compression [START_REF] Lloyd | Least squares quantization in PCM[END_REF] and is now more widely used in Machine Learning for unsupervised learning and clustering analysis, in numerical probability for (conditional) expectation computation and option pricing. More specifically, for µ some probability distribution in L 2 (R q ) and for fixed K ∈ N, the optimal quantization problem reads min x=(x 1 ,...,x K )∈(R q ) K V (x) := 1 2 R q min 1≤k≤K |ξ -x k | 2 µ(dξ).

(1.1.17)

Then defining the Voronoï partition of R q :

V k (x 1 , . . . , x K ) := ξ ∈ R q ∀ 1 ≤ j ≤ K, |ξ -

x k | ≤ |ξ -x j | , 1 ≤ k ≤ K,
we have

∂ x k V (x) = ∂ x k 1 2 K j=1 V j (x) |ξ -x j | 2 µ(dξ) = V k (x) (x k -ξ)µ(dξ) = E Y ∼µ 1 Y ∈V k (x) (x k -Y )
and the corresponding SGD algorithm, also called Competitive Learning Vector Quantization (CLVQ) in this case, reads with X n = (X 1 n , . . . , X K n ) ∈ (R q ) K and Y n ∼ µ and iid :

X n+1 = X n -γ n+1 1 Y n+1 ∈V k (Xn) (X k n -Y n+1 ) 1≤k≤K .
(1.1.18)

An example of optimal quantization using the CLVQ algorithm is given in Figure 1.6. We refer to [Pag15, Section 3.2] for more details about the CLVQ algorithm.

Stochastic processes and the Euler-Maruyama scheme

Stochastic differential equations

Let us consider a general SDE having its values in R d : where the number of quantizers is K = 5. We plot the density of µ, the points y k in blue, the centroids x k given by the CLVQ algorithm in red and the frontiers of the Voronoï partition (V k ) 1≤k≤5 in red.

dX t = b(X t )dt + σ(X t )dW t , X 0 ⊥ ⊥ W (1.1.19)
where W is a standard R q -valued Brownian motion and X 0 and W are both defined on some probability space (Ω, A, P). A method of approximate numerical simulation of X T for fixed time horizon T > 0 is the so-called Euler-Maruyama scheme which is the following time-discrete recursive algorithm: X0 = X 0 , Xn+1 = Xn + hb( Xn ) + h 1/2 σ( Xn )U n+1 , U n ∼ N (0, I d ) i.i.d.

(1.1. [START_REF]VBB +[END_REF] with h = T /N , N ∈ N and X T is approximated by XN . This scheme is used in particular for Monte Carlo estimation of expectations where for some f : R d → R, E[f (X T )] is approximated by the empirical mean M -1 M i=1 f ( X(i) N ), where X(i) N are M independent simulations following (1.1.20). Conversely, some discrete stochastic algorithms can be seen as the discretization of some SDE, as seen in the following sections.

In order to evaluate the quality of the approximation XN as N → ∞, we use strong error and weak error bounds. The strong error is defined for p ≥ 1 as the L p norm

E XN -X T p 1/p = ∥ XN -X T ∥ p .
Bounds on the strong error are obtained as follows. Assuming for example that b and σ are bounded and Lipschitz-continuous and writing XN = Y T with , where we used the Burkholder-Davis-Gundy and the regular and generalized Minkowski inequalities. But using the expression of the law of Y s conditionally to Y s, we have ∥Y s -Y s∥ p ≤ C(ss) 1/2 so that with φ t := sup s∈[0,t] ∥Y s -X s ∥ p we get

dY t = b(Y t)dt + σ(Y t)
φ t ≤ [b] Lip t 0 φ s ds + C BDG p [σ] Lip ( t 0 φ 2 s ds) 1/2 + Ch 1/2
but we have for every α > 0:

t 0 φ 2 s ds 1/2 ≤ φ 1/2 t t 0 φ s ds 1/2 ≤ α 2 φ s + 1 2α t 0 φ s ds and taking α = C BDG p [σ] Lip yields φ t ≤ C t 0 φ s ds + Ch 1/2 ,
where C is a constant depending on p, T , ∥b∥ ∞ , [b] Lip , ∥σ∥ ∞ and [b] Lip . By the Gronwall Lemma we obtain φ T ≤ Ch 1/2 for p ≥ 2; for ∈ [1, 2) the result still holds remarking that ∥ • ∥ p ≤ ∥ • ∥ 2 . Then the strong error ∥ XN -X T ∥ = ∥Y T -X T ∥ p is of order N -1/2 as N → ∞.

The weak error measures the error between the respective laws of X T and XN , denoted [X T ] and [ XN ] respectively, and is generally defined as the total variation distance:

d TV ([ XN ], [X T ]) = 2 sup A∈B(R d ) |P( XN ∈ A) -P(X T ∈ A)|,
or as the L p -Wasserstein distance:

W p ([ XN ], [X T ]) = inf
E[f ( XN )] -E[f (X T )], f ∈ A
where A is some class of Borel functions from R d to R; this last case corresponds to the total variation distance for A being the set of Borel measurable functions bounded by 1, and to the L 1 -Wasserstein distance for A being the set of Lipschitz-continuous functions with Lipschitz constant no greater than 1, according to the Monge-Kantorovich duality for W 1 .

The weak error is generally much more difficult to analyse than the strong error. For the L p -Wasserstein distance, the weak error can be directly bounded by the L p -strong error, however better bounds can be obtained. Talay and Tubaro [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF] and Bally and Talay [START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF] proved that the weak error is in general of order N -1 , against the order N -1/2 for the strong error, thus showing that the weak error fundamentally differs from the strong error and generally converges faster.

Stochastic Volterra equations

Let us consider (continuous) stochastic Volterra processes in R d i.e. which writes

X t = X 0 + t 0 K 1 (t, s)b(X s )ds + t 0 K 2 (t, s)σ(X s )dW s , t ∈ [0, T ],
(1.1.21)

where (W t ) is a standard Brownian motion in R q 3 , where b : R d → R q 1 , K 1 : [0, T ] 2 → M d,q 1 (R), σ : R d → M q 2 ,q 3 (R), K 2 : [0, T ] 2 → M d,q 2 (R), and where q 1 , q 2 , q 3 ∈ N. Stochastic Volterra equations (SVE) have been introduced for modelling in population dynamics, biology and physics [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Salah-Eldin | Stochastic differential systems with memory: Theory, examples and applications[END_REF], in order to generalize modelling to non-Markovian stochastic systems with some memory effect. They have been mathematically studied since [START_REF] Berger | Volterra equations with Itô integrals[END_REF] and [START_REF] Protter | Volterra equations driven by semimartingales[END_REF]. SVEs have recently attracted much attention in the mathematical finance community in the context of rough volatility modelling i.e. with K 1 (t, s) = K 2 (t, s) = (t -s) H-1/2 for some H ∈ (0, 1/2), yielding less regular trajectories than for classic SDEs because of the singularity of the kernel for s = t, and which are more able to reproduce some features of asset prices for small H, typically H ≃ 0.1 [ALV07, GJR18, EEFR18, JR20, Fuk17, Fuk21]. We recall that a rough stochastic volatility model is a special case of a singular two-dimensional SVE, where the first process is an asset price satisfying dS t = S t V t dB t for some Brownian motion B and where the second process (V t ) is the volatility satisfying some rough stochastic equation, then giving for the joint process (1.1.21) 2 × 2 matrix kernels K 1 and K 2 being diagonal and constant on their first coordinate. An approximation of the solution of (1.1.21) is given by the Euler-Maruyama scheme for SVEs:

Xn+1 = X 0 + h n j=0
K 1 ((n + 1)h, jh)b( Xj ) + h 1/2 n j=0 K 2 ((n + 1)h, jh)σ( Xj )(W t j+1 -W t j ) with h = T /N , N ∈ N and X T is approximated by XN .

The strong order of convergence of the Euler-Maruyama scheme for a rough SVE with parameter H is known to be N -H in general, which is very slow for small H ∈ (0, 1/2). The weak order rate is still an open problem, see [START_REF] Bayer | Weak error rates for option pricing under linear rough volatility[END_REF][START_REF] Bayer | Short communication: On the weak convergence rate in the discretization of rough volatility models[END_REF][START_REF] Gassiat | Weak error rates of numerical schemes for rough volatility[END_REF][START_REF] Peter | Weak error estimates for rough volatility models[END_REF] for recent advances.

Bayesian inference and sampling from probability measures

Stochastic algorithms are also used for sampling from a probability measure.

In Bayesian inference, we consider a family of probability distributions on R d 0 parametrized by a finite-dimensional parameter {p(u|x)du : x ∈ R d } and a prior distribution on the parameter p 0 (x)dx. Assuming that the observations u 1 , . . ., u M are i.i.d. and follow the distribution p(u|x)du conditionally to the value of the parameter x, the posterior probability on x ∈ R d knowing the observations has density proportional to p 0 (x)p(u 1 |x) . . . p(u M |x). Defining V (x) := -log(p 0 (x)p(u 1 |x) . . . p(u M |x)) = -log(p 0 (x)) -log(p(u

1 |x)) -• • • -log(p(u M |x)) =: V 0 (x) + V 1 (x) + • • • + V M (x),
then the posterior parameter X has a law of density proportional to e -V (x) . In this context, Bayesian inference requires sampling from a probability measure which is defined from a large amount of data if M is large.

Given a probability measure ν, Markov Chain Monte Carlo (MCMC) methods build an explicit Markov chain (X n ) admitting ν as its invariant measure, allowing to approximatively sample from ν. The Metropolis-Hastings algorithm was introduced by Metropolis in 1953 [MRR + 53] and extended by Hastings in 1970 [START_REF] Hastings | Monte Carlo sampling methods using Markov chains and their applications[END_REF]. Given a probability density ν and a parametrized Markov kernel chosen by the user Q(x, dy) on R d , the Markov chain X on R d is defined recursively by:

• Sample Xn+1 from Q(X n , dy).

• X n+1 = Xn+1 with acceptance probability min 1,

ν( Xn+1 )Q( Xn+1 , X n ) ν(X n )Q(X n , Xn+1 )
and X n+1 = X n otherwise so that the probability measure with density ν is the unique invariant distribution of X. One advantage of this algorithm is that it does not require to compute the normalization constant of the density ν nor the one of the kernel Q.

Lamberton and Pagès introduced in 2002 the method of sampling from a probability measure by solving SDEs [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]. That is, assuming that the SDE

dX t = b(X t )dt + σ(X t )dW t (1.1.22)
taking its values in R d is ergodic with invariant measure ν, then considering the Euler-Maruyama scheme with decreasing steps Xn+1 = Xn + γ n+1 b( Xn ) + √ γ n+1 σ( Xn )U n+1 , U n ∼ N (0, I d ) i.i.d., (1.1.23) the measure ν is estimated by the weighted average of Dirac measures

ν n := 1 Γ n n k=1 γ k δ Xk , Γ n = γ 1 + . . . + γ n .
(1.1.24) [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF] shows that for a general mean-reverting diffusion (1.1.22), the averaged Euler scheme of an ergodic diffusion converges to the invariant probability measure of the diffusion. More precisely:

Theorem 1.1.2. Assume that there exists a C 2 and coercive Lyapunov function

V : R d → [V ⋆ , ∞) such that ∇ 2 V is bounded and |∇V | 2 + |b| 2 ≤ CV, ∥σ(x)σ ⊤ (x)∥ = o(V (x)) as |x| → ∞,
(1.1.25) ∃α > 0, β ∈ R, ⟨∇V, b⟩ ≤ -αV + β, (1.1.26)

and that (1.1.22) admits a unique invariant measure ν. Then for every f : R d → R being continuous with f (x) = o(V k (x)) for some k ∈ N, then ν n (f ) → ν(f ) as n → ∞.

This method of simulation with respect to some probability measure fundamentally differs from using the Euler-Maruyama scheme for solving SDEs at fixed time horizon T > 0. It was then specifically analysed and extended in a series of papers, among them [LP03, Lem05, PP09, LP12, DM17, PR20].

Langevin equation and Langevin algorithms

More recently, [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] gives convergence rates for the law of Xn itself in the algorithm (1.1.23) for the L 1 -Wasserstein distance and the total variation distance. Assuming furthermore that σ is bounded and elliptic and that b is Lipschitz-continuous and the following confluence assumption:

∀x, y ∈ R d , W 1 ([X x t ], [X y t ]) ≤ C|x -y|e -ρt , (1.1.27)
which is fulfilled in particular if the uniform dissipative assumption outside some compact set is satisfied:

∃α 0 > 0, ∃R > 0, ∀x, y ∈ B(0, R), ⟨b(x) -b(y), x -y⟩ ≤ -α 0 |x -y| 2 , (1.1.28)

the authors obtain a convergence speed at rate "almost" γ n , which reads:

W 1 ([ Xx n ], ν) ≤ Cγ n | log(γ n )|ϑ(x), (1.1.29) ∀ε > 0, ∃C ϵ > 0, d TV ([ Xx n ], ν) ≤ C ε γ 1-ε n ϑ(x), (1.1.30) ϑ(x) := (1 + |x|) ∧ V 2 (x).
The strategy of proof relies on a domino strategy (telescopic sum) inspired by proofs of weak error expansion of discretization schemes of diffusion processes, see [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF][START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF]. For a function f : R d → R being either Lipschitz-continuous (for the L 1 -Wasserstein distance) or measurable bounded (for the total variation distance), the domino strategy consists in a step-by-step decomposition of the weak error to produce an upper bound as follows:

|Ef ( Xx n ) -Ef (X x Γn )| = | Pγ 1 • • • • • Pγn f (x) -P Γn f (x)| = n k=1 Pγ 1 • • • • • Pγ k-1 • ( Pγ k -P γ k ) • P Γn-Γ k f (x) ≤ n k=1 Pγ 1 • • • • • Pγ k-1 • ( Pγ k -P γ k ) • P Γn-Γ k f (x) ,
(1.1.31)

where P and P are the transition kernels associated to X and X respectively. Then three terms appear:

• The time discretization error, for large k, corresponding to the error between X and X with small time horizon and where the error is controlled by classic weak and strong bounds on the error of the Euler-Maruyama scheme.

• The first ergodic error, for small k, corresponding to the error between X and X with large time horizon. In this case, the ergodic properties of X also apply to X.

• The second ergodic error, corresponding to the distance between X and its invariant measure ν.

The case of the total variation distance is more difficult to deal with than the case of the L 1 -Wasserstein distance, since d TV ([ Xx t ], [X x t ]) is in general difficult to bound for small t > 0. Instead, [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] relies on Malliavin bounds using regularization properties of the semi-group.

Langevin equation and Langevin algorithms

The Langevin Equation

If the function V to be minimized in (1.1.1) is not convex, the gradient descent algorithm can be trapped into a local minimum which is not the global minimum, see Figure 1.7. Moreover, saddle points of V may slow down the convergence of the algorithm [DPG + 14]. The stochastic gradient descent (1.1.2), adding a noise to the classic gradient descent, helps escaping from such traps [START_REF] Lazarev | Convergence of stochastic approximation procedures in the case of several roots of a regression equation[END_REF][START_REF] Brandière | Les algorithmes stochastiques contournent-ils les pièges?[END_REF]. Another possibility is to add an exogenous noise to the gradient descent, which also adds regularization and stability to the algorithm. Considering (1.1.2) and adding a white noise yields:

Xn+1 = Xn -γ n+1 ∇V ( Xn ) + ζ n+1 + σ √ γ n+1 U n+1 , (1.2.1)
where (U n ) is i.i.d, U 1 ∼ N (0, I d ) and σ > 0. Let us point out the fundamental difference between the noise γ n+1 ζ n+1 associated to the stochastic approximation of the gradient and the exogenous white noise √ γ n+1 U n+1 , in particular since γ n → 0 the first one is of order γ n and the second one of order √ γ n . This algorithm, called Stochastic Gradient Langevin Dynamics (SGLD) was introduced by Welling and Teh in [START_REF] Welling | Bayesian Learning via Stochastic Gradient Langevin Dynamics[END_REF]. Langevin algorithms as (1.2.1) are inspired by stochastic analysis and stochastic differential equations. Indeed, the continuous version of (1.2.1) is

dX t = -∇V (X t )dt + σdW t , (1.2.2)
where W is a standard d-dimensional Brownian motion. Equation (1.2.2) was introduced by Langevin in 1908 [START_REF] Langevin | Sur la théorie du mouvement Brownien[END_REF] to model the random movement of a particle in a fluid colliding with the other particles. Assuming that exp(-2V /σ 2 ) ∈ L 1 (R d ) and denoting V ⋆ := min R d V , the Langevin equation (1.2.2) admits an invariant measure which is the Gibbs measure given by the density

ν σ (dx) := Z σ e -2(V (x)-V ⋆ )/σ 2 dx with Z σ := R d e -2(V (x)-V ⋆ )/σ 2 dx -1
.

(1.2.3) Indeed, for g : R d → R being C 2 with compact support, a quick calculation shows that

d dt E νσ [g(X t )] = d dt R d E x [g(X t )]ν σ (dx) = R d Lg(x)ν σ (dx) = Z σ R d -∇V (x) • ∇g(x) + σ 2 2 ∆g(x) e -2(V (x)-V ⋆ )/σ 2 dx = Z σ R d ∇ • ∇g(x)e -2(V (x)-V ⋆ )/σ 2 dx = 0, (1.2.4)
where we used an integration by parts formula to get the last line, where L is the infinitesimal generator associated to (1.2.2) and ∇• denotes the divergence operator. Moreover, for small σ, the measure ν σ is concentrated around the set argmin(V ). The following proposition makes this last statement precise. Proposition 1.2.1. Let V : R d → R be a Borel function such that

V ⋆ := essinf(V ) = inf{y : λ d {V ≤ y} > 0} > -∞.
Assume also that e -V ∈ L 1 (R d ) where λ d denotes the Lebesgue measure on R d . Then ∀ε > 0, ν σ ({V ≥ V ⋆ + ε}) → 0 as σ → 0.

Then if (X t ) defined in (1.2.2) is ergodic, solving (1.2.1) or (1.2.2) for small σ and large t guarantees that X t is close to argmin(V ) in some sense.

More precisely, convergence rates of Gibbs measures were studied by Hwang in 1980 [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF]. Assuming that argmin(V ) = {x ⋆ 1 , . . . , x ⋆ r } is finite and that each ∇ 2 V (x ⋆ i ), 1 ≤ i ≤ r, is positive definite then ν σ weakly converges to a mixture of Dirac measures at the x ⋆ i 's at rate σ. [START_REF] Hu | Mean-field Langevin dynamics and energy landscape of neural networks[END_REF] gives another heuristic for the Langevin equation and shows that some non-convex optimization problems can be seen as convex optimization problems when considering functionals defined on the infinite-dimensional space of probability measures on R d . The Brownian noise then appears as an entropic regularization term over the space of measures.

In the setting of Bayesian inference as presented in Section 1.1.4, [DM17, BDMS19, DM19, MMS20] study the method to sample from some target distribution ν defined on R d from the Langevin equation with constant noise coefficient σ ∈ R (1.2.2) [START_REF] Welling | Bayesian Learning via Stochastic Gradient Langevin Dynamics[END_REF]. Indeed, assuming that ν has positive probability density x → ν(x), then ν is the invariant measure of (1.2.2) with V (x) = -σ 2 2 log(ν(x)), ∇V (x) = -σ 2 2 ∇ν(x) ν(x) .

(1.2.5) [START_REF] Durmus | Nonasymptotic convergence analysis for the unadjusted Langevin algorithm[END_REF][START_REF] Durmus | High-dimensional Bayesian inference via the unadjusted Langevin algorithm[END_REF] give bounds for the total variation distance and the L 2 -Wasserstein distance between the discrete algorithm Xn in (1.2.1) and the target distribution ν. [START_REF] Mateusz | Nonasymptotic bounds for sampling algorithms without log-concavity[END_REF] focuses on the scheme with constant step size γ n ≡ γ > 0 and analyses the resulting asymptotic bias, giving bounds for the L 1 and L 2 -Wasserstein distances. As this is generally the case for MCMC methods, this method only requires the knowledge of the density of ν up to some normalization constant. These results are valid only for constant coefficient σ. In the additive setting, the exact diffusion and its Euler scheme are linked through the Girsanov formula so that the error is bounded using the Pinsker inequality [PP23, Appendix B]. However, this strategy cannot be applied to the multiplicative noise case, which turns out to be more demanding. [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF], which focuses on sampling from some distribution using (1.1.23), appears as an extension of [START_REF] Durmus | Nonasymptotic convergence analysis for the unadjusted Langevin algorithm[END_REF] to the case where σ is not constant.

Adaptive Langevin algorithms

Choosing in (1.2.1) and (1.2.2) a constant noise coefficient σ > 0, yielding an homogeneous and isotropic white noise σdW t may not be relevant in general. Indeed, the components of the gradient ∇V (X t ) may widely vary or may be correlated. To accelerate the convergence of X t to its invariant distribution, we rework (1.2.2) by allowing σ to depend on the position X t , leading to devise an adaptive Langevin algorithm. Allowing σ to be adaptive highly extends the range of applications of Langevin algorithms. The case where σ is not constant is called the "multiplicative case" whereas the case where σ is constant is called the 'additive case". A general heuristic is the so-called Newton method, which consists in considering σ = (∇ 2 V ) -1 and thus adding more noise in the directions where the gradient of V slowly varies in order to accelerate the exploration of the state space. However the exact computation of σ = (∇ 2 V ) -1 is cumbersome because • In high-dimensional optimization problems where the dimension d is large, the Hessian matrix has size d × d which is too large for both computation time and memory size.

• It becomes necessary to compute second-order derivatives whereas we only need to compute first-order derivatives for "classic" SGLD

• As the size d × d of the Hessian matrix is large, computing its inverse is also costly.

[DdVB15] adapted the well-known Newton method, which consists in considering σ = (∇ 2 V ) -1 , to SGLD. Since the size of the Hessian matrix may be too large in practice, because inverting it is computationally costly and because the Hessian matrix may not be positive in every point, it is suggested to consider instead |diag((∇ 2 V )) 2 | -1 . However, computing highorder derivatives may be cumbersome; [START_REF] Simsekli | Stochastic Quasi-Newton Langevin Monte Carlo[END_REF] adapts the quasi-Newton method [START_REF] Nocedal | Numerical optimization[END_REF] to approximate the Hessian matrix to SGLD, yielding the Stochastic Quasi-Newton Langevin algorithm.

[DHS11] and [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] give algorithms where the choice for σ is σ ≃ diag ((

λI d +[|∂ i V |] 1≤i≤d ) -1
), where λ > 0 guarantees numerical stability. The idea of using geometry has been explored in [START_REF] Patterson | Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex[END_REF], where σ -2 defines the local curvature of a Riemannian manifold, giving the Stochastic Gradient Riemaniann Langevin Dynamics algorithm where σ is equal to I -1/2 x where I x is the Fisher information matrix, or to some other choices (see [START_REF] Patterson | Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex[END_REF] Table 1]) as I x may be intractable. [START_REF] Ma | A Complete Recipe for Stochastic Gradient MCMC[END_REF] extends the previous algorithm to Hamiltonian Monte Carlo methods, where a momentum variable is added in order to take into account the "inertia" of the trajectory [START_REF] Radford | Monte Carlo Implementation[END_REF][START_REF] Durmus | Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo[END_REF], yielding the Stochastic Gradient Riemannian Hamiltonian Monte Carlo method.

Allowing the matrix σ to depend on the position yields a faster convergence; we refer to the previous references where the simulations prove that the new methods greatly improve classical stochastic gradients algorithms. In particular, we refer to the simulations [SBCR16, Figure 2], [PT13, Figure 2] and [MCF15, Figure 3] where the some of the above different methods are compared.

As presented in [START_REF] Ma | A Complete Recipe for Stochastic Gradient MCMC[END_REF] and [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF]Equation (3)] and formally demonstrated in [PP23, Proposition 2.6], if σ is not constant then we need to add a correction term to the drift so that the Gibbs measure ν σ remains the invariant measure.

Proposition 1.2.2. Let a > 0 and let (X t ) denote the solution to the R d -valued SDE in R d dX t = b(X t )dt + aσ(X t )dW t , X 0 ⊥ ⊥ W.

(1.2.6) then the distribution ν a defined in (1.2.3) is the unique invariant distribution of the above Brownian diffusion.

Assume that ∇ 2 V is bounded, e -2V /a 2 ∈ L 1 (R d ), σ
Defining the correction term Υ(x) := [ d j=1 ∂ j (σσ ⊤ )(x) ij ] 1≤i≤d which reduces to zero in the additive case, the discrete algorithm corresponding to the continuous SDE (1.2.6) reads

X n+1 = X n -γ k+1 σσ ⊤ (∇V (X n ) + ζ n+1 ) + a 2 γ n+1 Υ(X n ) + a √ γ n+1 σ(X n )U n+1 , (1.2.8)
where (U n ) n≥1 is i.i.d. and N (0, I d )-distributed.

Let us provide a simple example of how the above extension to non constant diffusion coefficients for Langevin simulation can be used, see [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF]Section 2.4]. We consider the pseudo-Cauchy distribution on R d with exponent κ > 0 defined by

π κ (dx) = C κ (1 + x 2 ) 1+κ dx = C κ e -V (x) dx with V (x) = (d + κ) log(1 + x 2 ) + 1.
Then using Proposition 1.2.2 with σ = I d , the distribution π κ is the invariant distribution of the one-dimensional Brownian diffusion

dY t = -(d + κ) Y t 1 + |Y | 2 t dt + dW t .
However, we have in this case |b(x)| → 0 as |x| → ∞, suggesting that the convergence to π κ is slow. On the other hand, using Proposition 1.2.2 with σ(x) = (1 + |x| 2 ) 1/2 I d , the measure π κ is also the invariant distribution of the Brownian diffusion

dX t = -(d + κ -1)X t dt + 1 + |X| 2 t dW t
where the drift satisfies ⟨b(x) -b(y), x -y⟩ = -(κ -1 + d/2)|x -y| 2 so that Assumption (1.1.28) is satisfied for κ > 1 -d/2, suggesting that the convergence to π κ is fast. For more details in particular concerning mean-reverting diffusions we refer to [PP23, Section 2.4].

Langevin-Simulated anneling equation

Considering (1.2.2) with invariant measure ν σ defined in (1.2.3) and since ν σ is concentrated around argmin(V ) for small σ (Proposition 1.2.1), a method for minimizing V is to solve (1.2.2) while making the drift coefficient decrease to zero along time, leading to the Langevin-simulated annealing equation:

dX t = -∇V (X t ) + a(t)σdW t , (1.2.9)
where X t is R d -valued, σ > 0 and a : R + → R + is decreasing and a(t) → 0 as t → ∞. Equation (1.2.9) is a non-homogeneous Markov SDE; for every t ∈ R + we can still define the "instantaneous" invariant measure as the invariant measure of the corresponding time-homogeneous SDE when freezing the time dependent coefficient a(t). In our case it is the measure ν a(t)σ . The Langevin-simulated annealing equation shares indeed its heuristic with the original simulated annealing algorithm [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF][START_REF] Van Laarhoven | Simulated annealing: theory and applications[END_REF], which builds a Markov chain from the Gibbs measure ν σ using the Metropolis-Hastings algorithm [MRR + 53] and where the parameter σ, interpreted as a temperature, slowly decreases to zero over the iterations. In Statistical Physics, the parameter sequence (a(t)σ) is interpreted as the square root of the system temperature. At the beginning of the algorithm, the temperature is high, allowing to explore the state space more efficiently to the detriment of the optimization procedure; at the end the temperature is low allowing to focus on the minima of V .

In [START_REF] Chiang | Diffusion for global optimization in R n[END_REF][START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF] is shown that choosing a(t) = A log -1/2 (t) for some A > 0 in (1.2.9) guarantees the convergence of X t to ν ⋆ defined as the limit measure of (ν a ) as a → 0, ν ⋆ being supported by argmin(V ). [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre[END_REF] proves again the convergence of the SDE using free energies inequalities. These studies deeply rely on some Poincaré and log-Sobolev inequalities and require the following assumptions on the potential function:

lim |x|→∞ V (x) = lim |x|→∞ |∇V (x)| = ∞ and ∀x ∈ R d , ∆V (x) ≤ C + |∇V (x)| 2 .
More specifically, taking σ = 1 in (1.2.9) and defining the Kullback-Liebler divergence (or free energy):

J t := d KL X t ∥ν a(t) = R d log p(t, x) ν a(t) (x) p(t, x)dx, (1.2.10)
where p(t, x) is the density of X t at x ∈ R d (see [Fri64, Chapter 9, Theorem 7]) and follows the Fokker-Planck equation:

∂ t p(t, x) = ∇ • (∇V (x)p(t, x)) + 1 2 a 2 (t)∆p(t, x), (1.2.11)
using the expression of ∂ t p(t, x) in (1.2.11), the explicit expression of the density ν a(t) (x) and the integration by parts formula, we obtain

dJ t dt = 2 a 4 (t) da 2 (t) dt R d V (x)(ν a(t) (x) -p(t, x))dx -2a 2 (t) R d ∇ p(t, x) ν a(t) (x) 2 ν a(t) (x)dx. (1.2.12)
We then use a logarithmic Sobolev inequality:

R d f 2 log(f 2 )dν a(t) ≤ C R d |∇f | 2 dν a(t) + R d f 2 dν a(t) log R d f 2 dν a(t) (1.2.13) with f (x) = p(t, x)/ν a(t) (x) so that -2σ 2 (t) R d ∇ p(t, x) ν a(t) (x) 2 ν a(t) (x)dx ≤ -C R d p(t, x) ν a(t) (x) log p(t, x) ν a(t) (x) ν a(t) (x)dx = -CJ t .
On the other hand,

R d V (x)(ν a(t) (x) -p(t, x))dx = ν a(t) (V ) -EV (X t )
can be bounded independently of t using classic bounds on the growth of the potential, see for example [PP23, Proposition A.1]. Then (1.2.12) gives a bound on dJ t /dt and solving the differential inequation with a(t) = A log -1/2 (t) yields the convergence of J t to 0.

[Zit08] proves that the convergence still holds under weaker assumptions, in particular where the gradient of the potential is not coercive, using weak Poincaré inequalities. In [START_REF] Saul | Recursive stochastic algorithms for global optimization in R d[END_REF] is proved the convergence of the associated stochastic gradient descent algorithm.

Contributions of the thesis

Our research results and articles are available on my personal research website at the address: https://perso.lpsm.paris/~pbras/.

Our code for simulations along with Jupyter notebooks are available on my GitHub page at the address:

https://github.com/Bras-P.

Convergence of Langevin-Simulated annealing algorithms

Convergence of the adaptive Langevin equation and its Euler-Maruyama scheme for the L 1 -Wasserstein distance and in total variation

In Chapters 2 and 3, we prove the convergence and give convergence rates for Langevin-simulated annealing algorithms with multiplicative noise i.e. for the SDE (1.2.6) and its associated Euler-Maruyama scheme (1.2.8) where the coefficient of the exogenous noise (a(t)) decreases to 0 as t → ∞. The target measure denoted ν ⋆ is defined as the limit measure of ν a as a → 0 and the rates of convergence are given for the L 1 -Wasserstein distance and the total variation distance.

We make assumptions on V and σ similar to those in [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF], in particular that |∇V | 2 ≤ CV , ∇V is Lipschitz continuous and σ is bounded with bounded derivatives. We also assume that argmin(V ) is finite and that for every

x ⋆ ∈ argmin(V ), either ∇ 2 V (x ⋆ ) is definite positive or x ⋆ is a strict polynomial minimum.
We adopt a domino strategy (see (1.1.31) and the description following) and use methods described in Section 1.1.3.1 and in Section 1.1.4 however we need to adapt the strategy to nonhomogeneous Markov diffusion processes with evanescent ellipticity, since the diffusion coefficient of the SDE depends on the time t and its ellipticity fades away as t → ∞. We prove that choosing a coefficient (a(t)) of order log -1/2 (t) is a sufficient -and generally necessary -condition for convergence. We give a convergence rate for W 1 ([X t ], ν ⋆ ) which turns out to be somehow limited by W 1 (ν a(t) , ν ⋆ ) which is of order a(t) ≃ log -1/2 (t) under the assumption that ∇ 2 V is positive definite at every point of argmin(V ). Still we establish sharper bounds for the convergence of W 1 (X t , ν a(t) ):

W 1 ([X x 0 t ], ν ⋆ ) + W 1 ([ Xx 0 N (t) ], ν ⋆ ) ≤ C max(1 + |x 0 |, V 2 (x 0 ))a(t), W 1 ([X x 0 t ], ν a(t) ) + W 1 ([ Xx 0 N (t) ], ν a(t) ) ≤ C α max(1 + |x 0 |, V 2 (x 0 ))t -α
for every α ∈ (0, 1) with N (t) = min{k ∈ N, Γ k+1 > t}, see Theorems 2.2.1 and 2.2.4. For the total variation distance, we obtain for every α ∈ (0, 1):

d TV ([X x 0 t ], ν a(t) ) ≤ C α e C log 1/2 (t)(1+|x 0 |) t -α , (1.3.1) d TV ([ Xx 0 N (t) ], ν a(t) ) ≤ C α log 1/2 (t) max(V 2 (x 0 ), 1 + |x 0 |)t -α + e C log 1/2 (t)(1+|x 0 | 2 ) t C/A 2 γ 1/2 N (Ct) (1.3.2)
with A defined by a(t) = A log -1/2 (e + t), see Theorem 3.2.1. These results have been submitted as two separate articles as joints work with Gilles Pagès:

• The analysis of the convergence for the L 1 -Wasserstein distance, entitled Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise and accepted for publication in Mathematics of Computation,

• The analysis of the convergence for the total variation distance, entitled Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise II: Total Variation and which has been published in Monte Carlo Methods and Applications.

Convergence rates of Gibbs measures

In order to analyse the rate of convergence of Langevin algorithms, for D some distance on P(R d ), let us write

D([X t ], ν ⋆ ) ≤ D([X t ], ν a ) + D(ν a , ν ⋆ ) if a > 0 is constant, D([X t ], ν a(t) ) + D(ν a(t) , ν ⋆ ) if a is decreasing to 0. (1.3.3)
The second term in this inequality D(ν a , ν ⋆ ) corresponds to the bias term. It measures the distance between the invariant distribution ν ⋆ and its approximation using a Gibbs measure ν a and converges to 0 as a → 0. Consequently we also need to investigate the order of D(ν a , ν ⋆ ) for small a > 0. It is known that if argmin(V ) is finite and if for every x ⋆ ∈ argmin(V ), ∇ 2 V (x ⋆ ) is definite positive then ν a converges weakly to ν ⋆ which is a weighted sum of Dirac masses where each weight is proportional to det -1/2 (∇ 2 V (x ⋆ )) and the rate of convergence is of order a as a → 0 [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF]. However, there are few results available when ∇ 2 V (x ⋆ ) is not positive definite for some x ⋆ ∈ argmin(V ). Assuming argmin(V ) = {x ⋆ } for simplifying notations, [START_REF] Athreya | Gibbs measures asymptotics[END_REF] gives a convergence rate under the condition that there exists a function g :

R d → R with e -g ∈ L 1 (R d ) and α 1 , . . . , α d ∈ (0, +∞) such that ∀h ∈ R d , 1 a [V (x ⋆ + (a α 1 h 1 , . . . , a α d h d )) -V (x ⋆ )] -→ a→0 g(h 1 , . . . , h d ). (1.3.4)
In Chapter 4 we give conditions on V such that (1.3.4) is fulfilled and then elucidate the expression of g depending on V and its successive derivatives at x ⋆ . Instead of positive definiteness we assume x ⋆ is a strictly polynomial minimum of V i.e. V -V (x ⋆ ) is bounded below in a neighbourhood of x ⋆ by some non-negative polynomial function null only at x ⋆ . Under some assumptions, we give an algorithm to identify α = (α 1 , . . . , α d ) ∈ R d , an orthogonal transformation B and g : R d → R a polynomial function which is not constant in any of its variables, depending on the successive derivatives of V such that

∀h ∈ R d , 1 a V (x ⋆ + B • (a α * h)) -V (x ⋆ ) -→ a→0 g(h)
where a α is the vector (a α 1 , . . . , a α d ) and * denotes the Schur product between two vectors i.e.

for u, v ∈ R d , u * v = (u 1 v 1 , . . . , u d v d ).
Then we obtain the following central limit theorem:

a -α * (B -1 • (Y a -x ⋆ )) -→ a→0 Y in law
where Y a ∼ ν √ 2a and Y ∼ e -g(y) dy. We refer to Theorems 4.3.3 and 4.3.5. Although the case where ∇ 2 V (x ⋆ ) is not positive definite may seem singular and mostly of theoretical interest at first sight, it can actually occur in practice as pointed out by eminent Machine Learning researchers, among them L. Bottou and Y. LeCun [SBL16, SEU + 17], for high-dimensional and over-parametrized optimization and inference problems, especially for the calibration of artificial neural networks.

These results have been published as the following article: Pierre Bras. Convergence rates of Gibbs measures with degenerate minimum. Bernoulli, 28(4):2431 -2458, 2022.

Adaptive Langevin algorithms for deep Neural Networks

SGLD algorithms for deep neural networks, Layer Langevin algorithm, application to image classification

In Chapter 5 we implement and run simulations to evaluate the performances of SGLD algorithms described in Section 1.2.2 with different choice for the function σ on various problems coming from Machine Learning: regression, classification, image recognition, time series analysis etc. In particular we analyse the benefits of adding Gaussian noise during the training in comparison with the non-Langevin counterpart algorithms. More precisely, popular preconditioned stochastic gradient methods in Machine Learning such as RMSprop [START_REF] Tieleman | Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude[END_REF], Adam and Adamax [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] and Adadelta [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] generally writes in a non-Langevin setting where g n+1 is an estimation of the gradient ∇V (X n ) and P n+1 is some preconditioner rule, while similarly to (1.2.8), the corresponding Langevin counterpart method writes

X n+1 = X n -γ n+1 P n+1 • g n+1 (1.3.5)
X n+1 = X n -γ n+1 P n+1 • g n+1 + aγ 1/2 n+1 N (0, P n+1 ) + a 2 γ n+1 d j=1 ∂ j P n+1 (X n ) ij 1≤i≤d . (1.3.6)
In Chapter 6 we proceed to a side by side comparison of (1. Still, as a price to be paid, deep neural networks are considerably more difficult to train [GB10, DPG + 14]. To cope with this issue, highway networks [START_REF] Rupesh | Training very deep networks[END_REF] and residual networks [START_REF] He | Deep residual learning for image recognition[END_REF] were introduced. Their many successive layers behaves either as a dense layer or as the identity function, allowing the gradient information to propagate trough the successive layers.

We compare the benefits of preconditioned Langevin algorithms [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] for various architectures and for various depths of neural networks and we observe that the deeper the network is, the greater are the gains provided by Langevin algorithms. Based on this heuristic and since the most important non-linearities of the network are contained in the deepest layers, we introduce a new optimization method that we call Layer Langevin algorithm, which consists in training the network by adding Langevin noise only to the training of some weights and not to the other weights. That is, the stochastic gradient method described in (1.3.6) becomes

X (i) n+1 = X (i) n -γ n+1 P n+1 • g n+1 (i) + 1 i∈J aγ 1/2 n+1 [N (0, P n+1 )] (i) + 1 i∈J a 2 γ n+1 d j=1 ∂ j P n+1 (X n ) ij (i) , 1 ≤ i ≤ d
for some J subset of {1, . . . , d}. In particular, we choose J to be the set of indexes of the weights of the k first (deepest) layers for some integer k. We refer to Section 6.4. We then highlight the possibilities of training acceleration using Langevin and Layer Langevin methods on deep residual [START_REF] He | Deep residual learning for image recognition[END_REF] and dense convolutional networks [START_REF] Huang | Densely connected convolutional networks[END_REF] for image classification.

We give an implementation of Langevin and Layer Langevin algorithms in TensorFlow as instances of the base class tf.keras.optimizers. Optimizers. These optimizers are directly

u • • • Y t k SDE Y t k+1 SDE Y t k+2 • • • G G G • • • + + + • • •

Langevin algorithms for deep stochastic control

In Chapter 7, we evaluate the benefits of Langevin algorithms on solving stochastic optimal control problems using neural networks as described in Section 1.1.2.3. Indeed, if the control is parametrized by a neural network and if it is applied at many discretization time steps, then the stochastic control problem reads as the training of a very deep neural network, see Figure 1.9.

As in Section 1.3.2.1, we compare preconditioned Langevin algorithms with their respective non-Langevin counterparts for solving various stochastic optimal control problems: fishing quotas [START_REF] Mathieu Laurière | Performance of a Markovian Neural Network versus dynamic programming on a fishing control problem[END_REF], deep hedging of financial options [START_REF] Buehler | Deep hedging[END_REF], oil drilling and resource management [START_REF] Gaïgi | Optimal risk management problem of natural resources: application to oil drilling[END_REF], and we show that Langevin and Layer Langevin optimizers can significantly improve the training procedure.

Simulation of stochastic processes and discretization schemes

Total variation convergence of the Euler-Maruyama scheme in small time

For the total variation bound in (1.3.2), while implementing the domino strategy described in Section 1.1.4 we need to establish a bound for the total variation between an SDE and its 1.3. Contributions of the thesis one-step Euler-Maruyama scheme in small time i.e. a bound for d TV ([ Xx t ], [X x t ]) where

X x 0 = x, dX t = b(X t )dt + σ(X t )dW t , Xt = x + tb(x) + σ(x)
W t where b and σ are general Lipschitz continuous coefficients here. Indeed, the difficulty of the total variation distance in small time is the following: considering its representation formula and comparing it with the L 1 -Wasserstein distance, if x and y ∈ R d are close to each other and if f : R d → R is Lipschitz continuous, then we can bound |f (x) -f (y)| by [f ] Lip |x -y|; whereas if f is simply measurable and bounded, then we cannot directly bound |f (x) -f (y)| in terms of |x -y|. Previous results in the literature only focus on total variation bounds for fixed time horizon T > 0 [START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF][START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF].

In Chapter 8, we show more generally that for two SDEs in R d starting at the same point:

X x 0 = x ∈ R d , dX x t = b 1 (X x t )dt + σ 1 (X x t )dW t , Y x 0 = x, dY x t = b 2 (Y x t )dt + σ 2 (Y x t )dW t and assuming that for i = 1, 2, σ i ∈ C 2 b and is elliptic and b ∈ C 1 with bounded derivatives, we obtain d TV ([X x t ], [Y x t ]) ≤ C(t 1/2 + |σ 1 -σ 2 |(x)) 2/3 + Ce c|x| 2 t 1/2 , (1.3.7)
see Theorem 8.2.1. This bound relies on the following result for r = 1 which links the total variation distance with the L 1 -Wasserstein distance provided that the laws of X x t and Y x t are regular enough: Theorem 1.3.1. Let Z 1 and Z 2 be two random vectors in L 1 (R d ) and admitting densities p 1 and p 2 respectively with respect to the Lebesgue measure. Assume furthermore that p 1 and p 2 are C 2r with r ∈ N and that ∇ k p i ∈ L 1 (R d ) for i = 1, 2 and k = 1, . . . , 2r. Then we have

d TV (Z 1 , Z 2 ) ≤ C d,r W 1 (Z 1 , Z 2 ) 2r/(2r+1) R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ 1/(2r+1) (1.3.8)
where the constant C d,r depends only on d and on r.

In the case of SDEs, the densities of X x t and Y x t can be expressed as the solution a Fokker-Planck partial differential equation and we rely on Aronson bounds [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] to control their regularity.

We furthermore extend (1.3.7): assuming that σ ∈ C 2r b we prove that

d TV ([X x t ], [Y x t ]) ≤ C(t 1/2 + |σ 1 -σ 2 |(x)) 2r/(2r+1) + Ce c|x| 2 t 1/2 , (1.3.9)
see Theorem 8.2.2. For r ≥ 2, we use a multi-step Richardson-Romberg extrapolation [START_REF] Fry | Ix. the approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam[END_REF][START_REF] Lemaire | Multilevel Richardson-Romberg extrapolation[END_REF] which is a method imported from numerical analysis and which builds a Taylor expansion with null coefficients up to some high order. These results have been published as: Pierre Bras, Gilles Pagès, and Fabien Panloup. Total variation distance between two diffusions in small time with unbounded drift: application to the Euler-Maruyama scheme. Electronic Journal of Probability, 27:1-19, 2022.

Weak error rates for numerical schemes of Stochastic Volterra Equations with application to stochastic volatility models and option pricing under path-dependent volatility

As a further research on convergence of Euler-Maruyama schemes for SDEs, in Chapter 9 we study the rate of weak convergence of numerical schemes for Stochastic Volterra Equations as described in Section 1.1.3.2, that is for f : R d → R being smooth we give bounds on

E[f ( XN )] -E[f (X T )].
(1.3.10)

A first bound on the weak error can be obtained from bounds on the strong error, however this is sub-optimal in general. For example, for Stochastic Differential Equations (SDE) the strong error is of order O(1/ √ N ) but the weak error is of order O(1/N ). Such bounds get even worse in the case of SVE with fractional kernel, giving a weak error bounded by the strong error which is order O(N -H ), where H ∈ (0, 1/2) is the Hurst parameter of the fractional kernel and is small (H ≃ 0.1) in many financial applications. In [START_REF] Richard | Discrete-time simulation of stochastic Volterra equations[END_REF] are given bounds for the weak error for the multi-level Euler-Maruyama scheme, however the authors only assume that the weak error is bounded by the strong error (see [RTY21, Section 2.3]), which is largely suboptimal in general. In [START_REF] Gassiat | Weak error rates of numerical schemes for rough volatility[END_REF] are given weak error rates from some rough volatility models and are proved to be of order O(N -(3H+1/2)∧1 ), yielding significantly better bounds in the case where H is close to 0. However the results are valid only for some special cases (semilinear or cubic test function), hinting that obtaining general results for fractional processes is difficult.

As a first step, we focus on the regular case i.e. where K 1 and K 2 are defined and continuous on [0, T ] 2 . Developing a domino strategy (1.1.31) that we adapt to the path-dependent case and using pathwise derivatives of path-dependent functionals [START_REF] Viens | A martingale approach for fractional Brownian motions and related path dependent PDEs[END_REF][START_REF] Dupire | Functional Itô calculus[END_REF], we prove that the convergence rate of (1.3.10) is of order O(1/N ) in the non-singular case and where N is the number of steps in the Euler-Maruyama scheme, see Theorem 9.2.2, thus giving a convergence rate similar to the Markovian SDE case.

These results are a joint work with Masaaki Fukasawa and have been submitted to SIAM Journal on Financial Mathematics (SIFIN ) and are currently in revision for possible publication. The analysis of the singular rough case from adapting from the regular case is a work in progress.

Simulation of reflected Brownian motion on two dimensional wedges

In Chapter 10 we study the Brownian motion W in R 2 which is reflected or stopped on a wedge, i.e. the subset of R 2 defined as

D = {(r cos(θ), r sin(θ)), r ≥ 0, θ ∈ [0, α]}
for some α ∈ (0, 2π) [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF][START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF][START_REF] Pilipenko | An introduction to stochastic differential equations with reflection[END_REF].

We prove the following density formula for the reflected Brownian motion X x t starting at x = (r 0 cos(θ 0 ), r 0 sin(θ 0 )):

P x (X t ∈ dy) = 2r tα e - r 2 +r 2 0 2t 1 2 I 0 rr 0 t + ∞ n=1 I nπ/α rr 0 t cos nπθ α cos nπθ 0 α drdθ (1.3.11)
where I a , a ≥ 0, stands for the modified Bessel function of the first kind, see Theorem 10.4.2. Unfortunately, this formula involves oscillating infinite sums of Bessel functions that are hardly usable for simulation purposes. Instead of directly computing these sums, we propose an alternative simulation method which uses an extension of the reflection principle in two dimensions for a particular type of wedges with angle π/m, m ∈ N in Sections 10.5.2 and 10.5.3. As a first step, we obtain a simulation method for E[f (W T ∧τ )] where τ stands for the first time the process W touches the boundary of the wedge D. Applying the methodology for the stopped process recursively one obtains an algorithm for the reflected process X. We then extend these methods to the simulation of the reflection on D of the process Z with dZ t = b(Z t )dt + dW t for some drift coefficient b.

The reflected Brownian motion and more generally reflected processes have applications in finance (for example, in stochastic models where the process, which can model an interest rates, is constrained to be non-negative [START_REF] Ha | Applications of the reflected Ornstein-Uhlenbeck process[END_REF] or has other constraints like barriers such as in [START_REF] Ichiba | Diffusions with rankbased characteristics and values in the nonnegative quadrant[END_REF]); in queueing models [START_REF] Giorno | On some diffusion approximations to queueing systems[END_REF], etc. Considering the particular case of a wedge may open the way to new simulation algorithms for reflected processes adapted to non-smooth domains with corner points.

These results have been published as the following article: Pierre Bras and Arturo Kohatsu-Higa. Simulation of reflected Brownian motion on two dimensional wedges. Stochastic Processes and their Applications, 156:349-378, 2023.

Part I

Convergence of adaptive Langevin-Simulated Annealing algorithms 2.1 Introduction

dX t = -∇V (X t )dt + σdW t , t ≥ 0,
where (W t ) is a d-dimensional Brownian motion and where σ > 0. Under standard assumptions, the invariant measure of this SDE is the Gibbs measure of density proportional to e -2V (x)/σ 2 and for small enough σ, this measure concentrates around argmin(V ), see [START_REF] Arnak | Theoretical guarantees for approximate sampling from smooth and log-concave densities[END_REF] and Chapter 4. Adding a small noise to the gradient descent allows to explore the space and to escape from traps such as local minima or saddle points appearing in non-convex optimization problems [Laz92, DPG + 14, HRSS21]. This noise may also be interpreted as coming from the approximation of the gradient in stochastic gradient descent algorithms. Such methods have been recently brought up to light again with Stochastic Gradient Langevin Dynamics (SGLD) algorithms [START_REF] Welling | Bayesian Learning via Stochastic Gradient Langevin Dynamics[END_REF][START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF], especially for the deep learning and the calibration of large artificial neural networks, which is a high-dimensional non-convex optimization problem. The Langevin-simulated annealing SDE is the Langevin equation where the noise parameter is slowly decreasing to 0, namely

dX t = -∇V (X t )dt + a(t)σdW t , t ≥ 0, (2.1.1)
where a : R + → R + is non-increasing and converges to 0. The idea is that the "instantaneous" invariant measure ν a(t)σ which is the Gibbs measure of density ∝ exp(-2V (x)/(a(t) 2 σ 2 )) converges itself to argmin(V ). This method indeed shares similarities with the original simulated annealing algorithm [START_REF] Van Laarhoven | Simulated annealing: theory and applications[END_REF], which builds a Markov chain from the Gibbs measure using the Metropolis-Hastings algorithm and where the parameter σ, interpreted as a temperature, slowly decreases to zero over the iterations.

In [START_REF] Chiang | Diffusion for global optimization in R n[END_REF][START_REF] Royer | A remark on simulated annealing of diffusion processes[END_REF] is shown that choosing a(t) = A log -1/2 (t) for some A > 0 in (2.1.1) guarantees the convergence of X t to ν ⋆ defined as the limit measure of (ν a(t) ) as t → ∞ and which is supported by argmin(V ). [START_REF] Miclo | Recuit simulé sur R n . Étude de l'évolution de l'énergie libre[END_REF] proves again the convergence of the SDE using free energies inequalities. These studies deeply rely on some Poincaré and log-Sobolev inequalities and require the following assumptions on the potential function:

lim |x|→∞ V (x) = lim |x|→∞ |∇V (x)| = ∞ and ∀x ∈ R d , ∆V (x) ≤ C + |∇V (x)| 2 .
[Zit08] proves that the convergence still holds under weaker assumptions, in particular where the gradient of the potential is not coercive, using weak Poincaré inequalities. In [START_REF] Saul | Recursive stochastic algorithms for global optimization in R d[END_REF] is proved the convergence of the associated stochastic gradient descent algorithm.

All these results are established in the so-called additive case, i.e. they highly rely on the fact that σ is constant, whereas little attention has been paid to the multiplicative case, i.e. where σ : R d → M d (R) is not constant and depends on X t . Allowing σ to be adaptive and to depend on the position highly extends the range of applications of Langevin algorithms and such adaptive algorithms are already widely used by practitioners and prove to be faster than non-adaptive algorithms and competitive with standard non-Langevin algorithms or even faster. See Section 2.3.2 where various specifications for σ(x) that can be found in the Stochastic Optimization literature are briefly presented, and Section 2.9 where we show results of simulations of the training of an artificial neural network for various choices of σ. However, to our knowledge, a general result of convergence for Langevin algorithms with multiplicative noise is yet to be proved. [PP23, Proposition 2.6] gives a general formula on b and σ so that the associated Gibbs measure is still the invariant measure of the SDE dX t = b(X t )dt + σ(X t )dW t ; a simple example of acceleration of convergence using non-constant σ is then given in [PP23, Section 2.4]. More generally, [START_REF] Ma | A Complete Recipe for Stochastic Gradient MCMC[END_REF] gives a characterization of any SGMCMC (Stochastic Gradient Markov Chain Monte Carlo) algorithm with multiplicative noise and with the corresponding Gibbs measure as a target. In practice, the matrix σ is often chosen so that σσ ⊤ ≃ (∇ 2 V ) -1 but approximations are needed because of the high dimensions of the matrix (e.g. only considering diagonal matrices). Still, our results hold also for non-diagonal σ, which opens the way to algorithms with such σ.

In this paper, we consider the following SDE:

dY t = -(σσ ⊤ ∇V )(Y t )dt + a(t)σ(Y t )dW t +   a 2 (t)   d j=1 ∂ j (σσ ⊤ )(Y t ) ij   1≤i≤d   dt (2.1.2) a(t) = A log(t) , (2.1.3)
where the expression of the drift comes from [PP23, Proposition 2.6] and where the second drift term is interpreted as a correction term so that ν a(t) is still the "instantaneous" invariant measure and satisfying -b a(t) ν + (1/2)∇ • (a(t)σν) = 0 for every t, where b a denotes the associated drift. This last term boils down to 0 if σ is constant. The aim of this paper is to prove the convergence for the L 1 -Wasserstein distance of the law of Y t to ν ⋆ in the setting adopted in [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF], assuming in particular the convex uniformity of the potential outside a compact set and the ellipticity and the boundedness of σ. We also prove the convergence of the corresponding Euler-Maruyama scheme with decreasing steps and with noisy measurements of the gradient arising from minibatch sampling. In this paper we use Markov models along with stochastic calculus to study the convergence of adaptive Langevin algorithms as a first theoretical study, however in practice these algorithms often include history/momentum effects with time-exponential average, and do not solely depend on Y t as assumed in (2.1.2). We refer to Section 5.2 for precise definitions of some popular gradient algorithms.

Considering the convex condition outside a compact set is in fact quite different from the convex setting and turns out to be more demanding. This setting often appears in optimization problems (see Section 2.3.1), where a characteristic set -the compact set -contains the interesting features of the model with traps such as local minima, and where outside of this set the loss function is coercive and convex. We give classic examples of neural networks where this setting applies.

We adopt a domino strategy like in [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF], inspired by proofs of weak error expansion of discretization schemes of diffusion processes [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF][START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF]. In [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] is proved the convergence of the Euler-Maruyama scheme X with decreasing steps (γ n ) of an ergodic and homogeneous SDE X with non constant σ, to the invariant measure of X; the additive case was tackled in [DM17, [START_REF] Brosse | The tamed unadjusted langevin algorithm[END_REF][START_REF] Durmus | High-dimensional Bayesian inference via the unadjusted Langevin algorithm[END_REF] and in [START_REF] Mateusz | Nonasymptotic bounds for sampling algorithms without log-concavity[END_REF] for the constant step case. It then appears that the multiplicative case is more demanding than the additive case. For a function f : R d → R, the domino strategy consists in a step-by-step decomposition of the weak error to produce an upper bound as follows:

|Ef ( Xx Γn ) -Ef (X x Γn )| = | Pγ 1 • • • • • Pγn f (x) -P Γn f (x)| ≤ n k=1 Pγ 1 • • • • • Pγ k-1 • ( Pγ k -P γ k ) • P Γn-Γ k f (x) , (2.1.4)
where P and P are the transition kernels associated to X and X respectively and where

Γ n = γ 1 + • • • + γ n .
Then two terms appear: first the "error" term, for large k, where the error is controlled by classic weak and strong bounds on the error of an Euler-Maruyama scheme, and the "ergodic" term, for small k, where the ergodicity of X is used. However, we cannot directly apply this strategy of proof to our problem since we consider a non homogeneous SDE Y , so we proceed as follows: we consider instead the SDE X where the coefficient a(t) is non-increasing and piecewise constant and where the successive plateaux [T n-1 , T n ) of a are increasingly larger time intervals. On each plateau we obtain a homogeneous and uniformly elliptic SDE with an invariant Gibbs distribution ν an where a n is the constant value of a on [T n-1 , T n ), to which a domino strategy can be applied. This ellipticity fades with time since a n goes to 0 and we need to carefully control its impact on the way the diffusion X gets close to its "instantaneous" invariant Gibbs distribution ν an . To this end we have to refine several one step weak error results from [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] and ergodic bounds from [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF], which extends the ergodic contraction bounds from [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF] to the multiplicative case. Doing so we derive by induction an upper-bound for the distance between X t and ν ⋆ after each plateau and prove that a coefficient (a(t)) of order log -1/2 (t) is a sufficient and generally necessary condition for convergence. Using this result, we then prove the convergence of Y t and its Euler-Maruyama scheme Ȳt by bounding the distance between X t and Y t and X t and Ȳt . We need to split the problem into the cases where (a(t)) is piecewise constant and where (a(t)) is continuously decaying and then use careful weak error comparisons to prove the convergence in the second case, as we cannot use ergodic properties for the non-homogeneous processes at first sight. We also consider the "Stochastic Gradient case" i.e. where the true gradient cannot be computed exactly and where a noise, which is a sequence of increments of a martingale, is added to the gradient. This case was treated in [START_REF] Saul | Recursive stochastic algorithms for global optimization in R d[END_REF] in the additive setting. The process X is used as a tool for the proof of the convergence of Y t , however the convergence of X t to ν ⋆ also has its own interest since the "plateau" method is also used by practitioners.

We also establish a convergence rate which is somehow limited by W 1 (ν a(t) , ν ⋆ ), which is of order a(t) under the assumption that argmin(V ) is finite and that ∇ 2 V is positive definite at every element of argmin(V ), thus giving the same convergence rate as in the additive case and with the same rate for the annealing schedule (a(t)), see Remarks 2.4.7 and 2.5.2. If argmin(V ) is still finite but if ∇ 2 V is not positive definite at every element of argmin(V ), but if we assume instead that all the elements of argmin(V ) are strictly polynomial minima, then the rate is of order a(t) δ for some δ ∈ (0, 1), see Chapter 4. We pay particular attention to the non-definite case, since it was pointed out in [SBL16, SEU + 17] that for some optimization problems arising in Machine Learning, the Hessian of the loss function at the end of the training tends to be extremely singular. Indeed, as the dimension of the parameter which is used to minimize the loss function is large and as the neural network can be over-parametrized, many eigenvalues of the Hessian matrix are close to zero. However, this subject is still new in the Stochastic Optimization literature and needs more theoretical background.

Still we give sharper bounds on the rate of convergence of the L 1 -Wasserstein distance between X or Y and ν a(t) as in practice the optimization procedure stops at some (large) t and the target distribution is actually ν a(t) instead of ν ⋆ .

In a next paper, we shall prove the convergence in total variation distance. In this last case, the domino strategy is more complex to implement and requires regularization lemmas, as in [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] which studies the convergence for both distances.

The article is organized as follows. In Section 2.2 we first give the setting and assumptions of the problem we consider. This setting is taken from [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF]. We then state our main results of convergence as well as convergence rates. In Section 2.3 we show how this setting applies to some classic optimization problems arising in Machine Learning and present several general choices for σ that are used in practice. In Section 2.4 we consider the case where the coefficient a is constant and give convergence rates to the invariant measure taking into account the ellipticity parameter. We also give preliminary lemmas for the rate of convergence of ν an to ν ⋆ . In Section 2.5 we prove the convergence of the solution of the SDE where a is piecewise constant, by "plateaux". Using the dependence in a of the rate of convergence to the invariant measure in the ergodic case we prove the convergence to argmin(V ). In Section 2.6 we prove the convergence of the SDE in the case where a is not by plateau but is continuously decreasing. This is done by bounding the Wasserstein distance with the "plateau" case and revisiting the lemmas for strong and weak errors from [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF]. In Section 2.7 and Section 2.8 we also prove the convergence for the corresponding Euler-Maruyama schemes. The proofs actually follow the same strategy as the previous one. In Section 2.9 we present experiments of training of neural networks using various specifications for σ; the algorithms with multiplicative σ prove to be faster than the algorithm with constant σ.

Notations

In addition to the notations given from page 1, we use the notation C to denote a positive real constant, which may change from line to line. The constant C depends on the parameters of the problem: the coefficients of the SDE, the choice of A in a(t) = A log -1/2 (t), the upper bound γ on the decreasing steps, but C does not depend on t nor x.

Assumptions and main results

Assumptions

Let V : R d → (0, +∞) be a C 2 potential function such that V is coercive and

(x → |x| 2 e -2V (x)/A 2 ) ∈ L 1 (R d ) for some A > 0.
(2.2.1)

Then V admits a minimum on R d . Moreover, let us assume that

V ⋆ := min R d V > 0, argmin(V ) = {x ⋆ 1 , . . . , x ⋆ m ⋆ }, ∀ i = 1, . . . , m ⋆ , ∇ 2 V (x ⋆ i ) > 0, (2.2.2, H V 1 )
i.e. min R d V is attained at a finite number m ⋆ of points and in each point the Hessian matrix is positive definite. We then define for a ∈ (0, A] the Gibbs measure ν a of density :

ν a (dx) = Z a e -2(V (x)-V ⋆ )/a 2 dx, Z a = R d e -2(V (x)-V ⋆ )/a 2 dx -1 (2.2.3)
We use the previous normalization for ν a and Z a however we emphasize that computing the density ν a (x) does not require the knowledge of V ⋆ since we could write ν a (dx) = Za exp(-2V (x)/a 2 )dx with the appropriate normalization constant Za . Following [Hwa80, Theorem 2.1], the measure ν a converges weakly to ν ⋆ as a → 0, where ν ⋆ is the weighted sum of Dirac measures:

ν ⋆ =   m ⋆ j=1 det ∇ 2 V (x ⋆ j ) -1/2   -1 m ⋆ i=1 det ∇ 2 V (x ⋆ i ) -1/2 δ x ⋆ i . (2.2.4)
We consider the following Langevin SDE in R d :

Y x 0 0 = x 0 ∈ R d , (2.2.5) dY x 0 t = b a(t) (Y x 0 t )dt + a(t)σ(Y x 0 t )dW t ,
where, for a ≥ 0, the drift b a is given by

b a (x) = -(σσ ⊤ ∇V )(x) + a 2   d j=1 ∂ j (σσ ⊤ ) ij (x)   1≤i≤d =: -(σσ ⊤ ∇V )(x) + a 2 Υ(x), (2.2.6)
where W is a standard R d -valued Brownian motion defined on some rich enough probability space (Ω, A, P), where σ :

R d → M d (R) is C 2 and a(t) = A log(t + e) (2.2.7)
where A is defined in (2.2.1) and with log(e) = 1. This equation corresponds to a gradient descent on the potential V with preconditioning σ and multiplicative noise; the second term in the drift (2.2.6) is a correction term (see [PP23, Proposition 2.6]) which is zero for constant σ.

We make the following assumptions on the potential V :

lim |x|→+∞ V (x) = +∞, |∇V | 2 ≤ CV and sup x∈R d ||∇ 2 V (x)|| < +∞, (2.2.8, H V 2 )
which implies in particular that V has at most a quadratic growth. Let us also assume that σ is bounded and Lipschitz continuous,

∇ 2 σ is bounded, ∇(σσ ⊤ )∇V is bounded, (2.2.9, H σ )
and that σ is uniformly elliptic, i.e.

∃ σ0 > 0, ∀x ∈ R d , (σσ ⊤ )(x) ≥ σ2 0 I d . ( 2
∈ [0, A], b a is [b] Lip -Lipschitz continuous.
We make the non-uniform dissipative (or convexity) assumption outside of a compact set: there exists α 0 > 0 and R 0 > 0 such that

∀x, y ∈ B(0, R 0 ) c , σσ ⊤ ∇V (x) -σσ ⊤ ∇V (y), x -y ≥ α 0 |x -y| 2 . (2.2.11, H cf )
Taking y ∈ B(0, R 0 ) c fixed, letting |x| → ∞ and using the boundedness of σ, (2.2.11, H cf ) implies that |∇V | is coercive. Using (2.2.8, H V 2 ) and the boundedness of σ, there exists C > 0 (depending on A) such that:

∀a ∈ [0, A], 1 + |b a (x)| ≤ CV 1/2 (x).
Let (γ n ) n≥1 be a non-increasing sequence of varying positive steps. We define Γ n := γ 1 + • • • + γ n and for t ≥ 0:

N (t) := min{k ≥ 0 : Γ k+1 > t} = max{k ≥ 0 : Γ k ≤ t}.
We make the classical assumptions on the step sequence, namely

γ n ↓ 0, n≥1 γ n = +∞ and n≥1 γ 2 n < +∞ (2.2.12, H γ1 )
and we also assume that

ϖ := lim sup n→∞ γ n -γ n+1 γ 2 n+1 < ∞. (2.2.13, H γ2 ) For example, if γ n = γ 1 /n α with α ∈ (1/2, 1) then ϖ = 0; if γ n = γ 1 /n then ϖ = γ 1 .
In Stochastic Gradient algorithms, the true gradient is estimated using mini-batches of randomly selected training data, which introduces noise in the gradient estimates. This noise is modelled by a zero-mean random vector ζ, which law only depends on the current position. That is, let us consider a family of random fields (ζ n (x)) x∈R d ,n∈N such that for every n ∈ N,

(ω, x) ∈ Ω × R d → ζ n (x, ω
) is measurable and for all x ∈ R d , the law of ζ n (x) only depends on x and (ζ n (x)) n∈N is an i.i.d. sequence independent of W . We make the following assumptions:

∀x ∈ R d , ∀p ≥ 1, E[ζ 1 (x)] = 0 and E[|ζ 1 (x)| p ] ≤ C p V p/2 (x).
(2.2.14)

We then consider the Euler-Maruyama scheme with decreasing steps associated to (Y t ):

Ȳ x 0 0 = x 0 , Ȳ x 0 Γ n+1 = ȲΓn + γ n+1 b a(Γn) ( Ȳ x 0 Γn ) + ζ n+1 ( Ȳ x 0 Γn ) (2.2.15) + a(Γ n )σ( Ȳ x 0 Γn )(W Γ n+1 -W Γn ),
We extend Ȳ x 0

•

on R + by considering its genuine continuous interpolation:

∀t ∈ [Γ n , Γ n+1 ), Ȳ x 0 t = Ȳ x 0 Γn + (t -Γ n ) b a(Γn) ( Ȳ x 0 Γn ) + ζ n+1 ( Ȳ x 0 Γn ) (2.2.16) + a(Γ n )σ( Ȳ x 0 Γn )(W t -W Γn ).
Assumption (2.2.11, H cf ) along with the fact that b a is Lipschitz-continuous guarantee the noexplosion of the above processes and numerical schemes [START_REF] Talay | Second-order discretization schemes of stochastic differential systems for the computation of the invariant law[END_REF][START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF], see also Lemmas 2.6.1 and 2.7.1.

Main results

We now state our main results.

∈ R d , W 1 ([Y x 0 t ], ν ⋆ ) -→ t→∞ 0.
More precisely, for every t > 0:

W 1 ([Y x 0 t ], ν ⋆ ) ≤ C max(1 + |x 0 |, V (x 0 ))a(t)
and for every α ∈ (0, 1) we have

W 1 ([Y x 0 t ], ν a(t) ) ≤ C max(1 + |x 0 |, V (x 0 ))t -α . (b) Let Ȳ be
H γ2 ), that V is C 3 with ∥∇ 3 V ∥ ≤ CV 1/2 and that σ is C 3 with ∥∇ 3 (σσ ⊤ )∥ ≤ CV 1/2 . If A is large enough then for every x 0 ∈ R d , W 1 ([ Ȳ x 0 t ], ν ⋆ ) -→ t→∞ 0.
More precisely, for every t > 0:

W 1 ([ Ȳ x 0 t ], ν ⋆ ) ≤ C max(1 + |x 0 |, V 2 (x 0 ))a(t),
and for every α ∈ (0, 1) we have

W 1 ([ Ȳ x 0 t ], ν a(t) ) ≤ C max(1 + |x 0 |, V 2 (x 0 ))t -α .
Remark 2.2.2. In particular, if argmin V = {x ⋆ } is reduced to a point, we can rewrite the conclusions of Theorem 2.2.1 as ∥Y x 0 t -x ⋆ ∥ 1 → 0 and ∥ Ȳ x 0 t -x ⋆ ∥ 1 → 0 respectively and so on.

The degenerate case

In this subsection we consider the case where some of the ∇ 2 V (x ⋆ i )'s may be not definite positive but where the x ⋆ i 's are strictly polynomial minima, i.e. is V (x) -V (x ⋆ i ) is bounded below in a neighbourhood of x ⋆ i by a non-negative polynomial function null only in x ⋆ i . This case can be treated in a similar way using the change of variable given in Chapter 4.

First, let us restate the results from Theorem 4.3.3. To simplify, let us assume that argmin(V ) is reduced to a point.

Theorem 2.2.3. Assume that V is C 2p with p ≥ 2, is coercive, that argmin(V ) = {x ⋆ }, that e -AV ∈ L 1 (R d )
for some A > 0 and that x ⋆ is a strictly polynomial minimum of order 2p i.e. p is the smallest integer such that

∃r > 0, ∀h ∈ B(0, r) \ {0}, 2p k=2 1 k! ∇ k V (x ⋆ ) • h ⊗k > 0.
Assume also the technical hypothesis (4.3.4) if p ≥ 5. Then there exist B ∈ O d (R), α 1 , . . . , α d ∈ {1/2, . . . , 1/(2p)} and a polynomial function g : R d → R which is not constant in any of its variables such that

∀h ∈ R d , 1 s [V (x ⋆ + B • (s α 1 h 1 , . . . , s α d h d )) -V (x ⋆ )] -→ s→0 g(h).
Moreover assume that g is coercive. Then if

Z s ∼ ν √ 2s , (B -1 • (Z s -x ⋆ )) 1 s α 1 , • • • , (B -1 • (Z s -x ⋆ )) d s α d L -→ Z as s → 0,
where Z has density proportional to exp(-g).

H γ2 ), that V is C 3 with ∥∇ 3 V ∥ ≤ CV 1/2 and that σ is C 3 with ∥∇ 3 (σσ ⊤ )∥ ≤ CV 1/2
. Let us denote α min := min(α 1 , . . . , α d ). Then for every α ∈ (0, 1) we have

W 1 ([Y x 0 t ], ν a(t) ) ≤ C max(1 + |x 0 |, V (x 0 ))t -α , W 1 ([ Ȳ x 0 t ], ν a(t) ) ≤ C max(1 + |x 0 |, V 2 (x 0 ))t -α , W 1 ([Y x 0 t ], ν ⋆ ) ≤ C max(1 + |x 0 |, V (x 0 ))a(t) 2α min , W 1 ([ Ȳ x 0 t ], ν ⋆ ) ≤ C max(1 + |x 0 |, V 2 (x 0 ))a(t) 2α min .
The proof is given in the Supplementary Material.

Application to optimization problems

Potential function associated to a Neural Regression Problem

The setting described in Section 2.2 can first be applied to convex optimization problems where the potential function V has a quadratic growth as |x| → ∞. Classical examples are least-squares regression and logistic regression with quadratic regularization, that is:

min x∈R d 1 M M i=1 log(1 + e -v i ⟨u i ,x⟩ ) + λ 2 |x| 2 ,
where v i ∈ {-1, +1} and u i ∈ R d are the data samples associated with a binary classification problem and where λ > 0 is the regularization parameter.

We now consider a scalar regression problem with a fully connected neural network with quadratic regularization. Let φ : R → R be the sigmoid function. To simplify the proofs, we may consider instead a smooth function approximating the sigmoid function such that φ ′ has compact support. Let K ∈ N be the number of layers and for k = 1, . . ., K, let d k ∈ N be the size of the k th layer with

d K = 1. For u ∈ R d k-1 and for θ ∈ M d k ,d k-1 (R), we define φ θ (u) := [φ([θ • u] i )] 1≤i≤d k . The output of the neural network is ψ : R d 1 ,d 0 × • • • × R d K ,d K-1 × R d 0 → R ψ(θ 1 , . . . , θ K , u) = ψ(θ, u) = φ θ K • . . . • φ θ 1 (u). Let u i ∈ R d 0 and v i ∈ R be the data samples for 1 ≤ i ≤ M . The objective is minimize θ 1 ,...,θ K V (θ) := 1 2M M i=1 (ψ(θ 1 , . . . , θ K , u i ) -v i ) 2 + λ 2 |θ| 2 ,
where θ = (θ 1 , . . . , θ K ) and where λ > 0.

Proposition 2.3.1. Consider a neural network with a single layer : ψ(θ, u) = φ(⟨θ, u⟩). Assume that the data u and v are bounded, that u admits a continuous density and φ ′ has bounded support in R. Then V satisfies (2.2.8, H V 2 ) and for some R 0 , α 0 > 0,

∀x, y ∈ B(0, R 0 ) c , ⟨∇V (x) -∇V (y), x -y⟩ ≥ α 0 |x -y| 2 (2.3.1)
Proof. Let us define the measure Q on R d 0 ×R associated to the data, i.e. Q = (1/M ) M i=1 δ u i ,v i . Note that φ, φ ′ and φ ′′ are bounded. The function ψ is bounded so

2V (θ) = (φ(⟨θ, u⟩) -v) 2 Q(du, dv) + λ|θ| 2 ∼ λ|θ| 2 as |θ| → ∞, so V is coercive. Moreover, we have ∇V = uφ ′ (⟨θ, u⟩)(φ(⟨θ, u⟩) -v)Q(du, dv) + λθ so ∇V (θ) ∼ λθ as |θ| → ∞ and |∇V | 2 ≤ CV . Then, let us assume that the support of φ ′ is included in [-R, R], that u has its values in B(0, R) and v in [-R, R].
Then the set {u ∈ B(0, R), |⟨θ, u⟩| < R} has Lebesgue measure no larger than C/|θ| so ∇ 2 (φ(⟨θ, u⟩) -v) 2 Q(du, dv) ≤ C/|θ|, so outside the compact set {|θ| ≤ 2C/λ}, we have ∥∇ 2 V ∥ ≥ λ/2 which guarantees (2.3.1). This proposition can be extended to the case where φ is of sigmoid type, with fast decay of φ ′ as ±∞. However, we cannot directly extend this proposition to multi-layers neural networks. Nevertheless, if we consider that the training stops if a parameter becomes too large and if we replace ψ(θ, u) by ψ(ϕ(θ), u) where ϕ : R → R is a smooth approximation of x → min(x, R)1 x≥0 + max(x, -R)1 x<0 where R > 0 is large and where ϕ is applied in order to avoid over-fitting coordinate by coordinate then the resulting potential V with quadratic regularization satisfies (2.2.8, H V 2 ) and (2.3.1).

Practitioner's corner: choices for σ

In this section we briefly present general choices for the non-constant matrix σ that are often used in the Stochastic Optimization and Machine Learning literature.

[WT11] introduced the Stochastic Gradient Langevin Dynamics (SGLD) with constant preconditioner matrix σ. [START_REF] Dauphin | Equilibrated adaptive learning rates for non-convex optimization[END_REF] adapted the well-known Newton method, which consists in considering σσ ⊤ = (∇ 2 V ) -1 , to SGLD. Since the size of the Hessian matrix may be too large in practice, because inverting it is computationally costly and because the Hessian matrix may not be positive in every point, it is suggested to consider instead |diag((∇ 2 V )) 2 | -1/2 . However, computing high-order derivatives may be cumbersome; [START_REF] Simsekli | Stochastic Quasi-Newton Langevin Monte Carlo[END_REF] adapts the quasi-Newton method [START_REF] Nocedal | Numerical optimization[END_REF] to approximate the Hessian matrix to SGLD, yielding the Stochastic Quasi-Newton Langevin algorithm.

[DHS11] and [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] give algorithms where the choice for σ is σ ≃ diag((λ + |∇V |) -1 ), where λ > 0 guarantees numerical stability. The idea of using geometry has been explored in [START_REF] Patterson | Stochastic Gradient Riemannian Langevin Dynamics on the Probability Simplex[END_REF], where σ -2 defines the local curvature of a Riemannian manifold, giving the Stochastic Gradient Riemaniann Langevin Dynamics algorithm where σ is equal to I -1/2 x where I x is the Fisher information matrix, or to some other choices (see [PT13, Table 1]) as I x may be intractable. [START_REF] Ma | A Complete Recipe for Stochastic Gradient MCMC[END_REF] extends the previous algorithm to Hamiltonian Monte Carlo methods, where a momentum variable is added in order to take into account the "inertia" of the trajectory [START_REF] Radford | Monte Carlo Implementation[END_REF][START_REF] Durmus | Irreducibility and geometric ergodicity of Hamiltonian Monte Carlo[END_REF], yielding the Stochastic Gradient Riemannian Hamiltonian Monte Carlo method.

Allowing the matrix σ to depend on the position yields a faster convergence; we refer to the previous references where various simulations confirm that these new methods greatly improve classical stochastic gradients algorithms. In particular, we refer to the simulations [SBCR16, Figure 2], [PT13, Figure 2] and [MCF15, Figure 3] where the different methods based on multiplicative noise are compared.

In popular choices for σ like RMSprop [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] and Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF], the order of growth of σ is roughly

σ(x) ≃ diag η 1 (η 2 + U (x)) -1/4 , with η 1 , η 2 > 0 and U (x) = ∇V (x) * 2
where * denotes the component-wise product. Then σ is bounded and we have ∇U = 2∇V * ∇ 2 V and

∇σ = -(η 1 /2) η 2 + U -5/4 * ∇V * ∇ 2 V,
where the shapes of the tensors are broadcast with * if the orders are different. We obtain then that ∇σ is bounded. Moreover

∇(σσ) ⊤ • ∇V = -η 1 (η 2 + U ) -3/2 * ∇V * ∇ 2 V • ∇V is also bounded. Moreover ∇ 2 σ = (5/4)η 1 (η 2 + U ) -9/4 * ∇V * ∇ 2 V ⊗ 2 ∇V * ∇ 2 V
with the rule a ⊗ 2 b = (a ij b ik ) ijk , so that ∇ 2 σ is also bounded. Thus the assumptions on the growth of σ and on its derivatives (2.2.9, H σ ) are satisfied.

In practice, the annealing schedule (a(t)) converges slowly to 0 and practitioners often use adaptive Langevin algorithms with a being constant or decreasing on a few plateau values instead. At the end of the optimization procedure, the coefficient a(t) may thus not be zero, yielding a convergence to the Gibbs measure ν a(t) , which still gives a good approximation of argmin(V ) if a(t) is small enough and the theoretical bounds we give for the "plateau" case (see Theorems 2.5.1 and 2.8.1) can still be applied for fixed t ∈ R. Moreover, adding small Langevin noise can improve the learning procedure in comparison with non-Langevin gradient descent algorithms, even if a is taken constant [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF].

Langevin equation with constant time coefficient

In this section, we consider the following R d -valued homogeneous SDE:

X x 0 = x ∈ R d , dX x t = b a (X x t )dt + aσ(X x t )dW t , (2.4.1)
with a ∈ (0, A] and where b a is defined in (2.2.6). The drift is specified in such a way that the Gibbs measure ν a defined in (2.2.3) is the unique invariant distribution of (X x t ) (see [PP23, Proposition 2.6]).

Exponential contraction property

We now prove contraction properties of the SDE (2.4.1) under the uniform convex setting on the whole R d or outside a compact set (2.2.11, H cf ). If the uniform dissipative assumption holds on R d then we have the following contraction property.

Proposition 2.4.1. Let Z be the solution of

Z x 0 = x ∈ R d , dZ x t = b Z (Z x t )dt + σ Z (Z x t )dW t ,
where the coefficients b Z and σ Z are (globally) Lipschitz continuous. Assume the uniform convexity i.e. there exists α > 0 such that

∀x, y ∈ R d , ⟨b Z (x) -b Z (y), x -y⟩ + 1 2 ||σ Z (x) -σ Z (y)|| 2 ≤ -α|x -y| 2 .
(2.4.2)

Then: ∀x, y ∈ R d , W 1 ([Z x t ] , [Z y t ]) ≤ C|x -y|e -αt .
Proof. By the Itō lemma, t → e 2αt |Z x t -Z y t | 2 is a super-martingale, so

E |Z x t -Z y t | 2 ≤ e -2αt |x -y| 2 ,
which yields the desired result.

This proposition can be applied to X under the assumption

∀x, y ∈ R d , ⟨b a (x) -b a (y), x -y⟩ + a 2 2 ||σ(x) -σ(y)|| 2 ≤ -α|x -y| 2 ,
which may be hard to check because of the dependence in a. In [PP23, Corollary 2.5] is proved that this contraction property is still true under uniform convexity outside a compact set (2.2.11, H cf ) using bounds from [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF]. [Wan20, Theorem 2.6] relies on explicit couplings of [X x t ] and [X y t ] by reflection. Such reflection couplings were introduced in [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF] for the additive case σ = I d and were extended to the multiplicative case in [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF]. In the following theorem we make explicit the dependence in a.

Theorem 2.4.2. Under the assumption (2.2.11, H cf ),

(a) For every x, y ∈ R d , W 1 ([X x t ] , [X y t ]) ≤ Ce C 1 /a 2 |x -y|e -ρat , ρ a := e -C 2 /a 2 (2.4.3, P cf )
where the constants C, C 1 , C 2 do not depend on a.

(b) For every x ∈ R d , W 1 ([X x t ] , ν a ) ≤ Ce C 1 /a 2 e -ρat W 1 (δ x , ν a ).
(2.4.4)

Proof. (a) We refine the proof of [Wan20, Theorem 2.6] to enhance the dependence of the constants in the parameter a. First we remark as in [PP23, Section 4.5] in the proof of Corollary 2.5, that Assumption (2.17) of [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF], stating that there exist constants K 1 , K 2 and r 0 > 0 such that, with σ :

= σσ ⊤ -σ2 0 I d , a 2 2 ∥ σ(x) -σ(y)∥ 2 - |a 2 (σ(x) -σ(y)) ⊤ (x -y)| 2 2|x -y| 2 + ⟨b a (x) -b a (y), x -y⟩ ≤ (K 1 + K 2 )1 |x-y|≤r 0 -K 2 |x -y| 2 , x, y ∈ R d , (2.4.5)
is true, since a ∈ (0, A] and σσ ⊤ bounded, as soon as there exist positive constants

K 1 , K 2 and R 1 such that ∀x, y ∈ R d , ⟨b a (x) -b a (y), x -y⟩ ≤ K 1 1 |x-y|≤R 1 -K 2 |x -y| 2 ,
which is, up to changing the positive constants, equivalent to

∀x, y ∈ R d , ⟨b 0 (x) -b 0 (y), x -y⟩ ≤ K 1 1 |x-y|≤R 1 -K 2 |x -y| 2 ,
which is in turn equivalent to (2.2.11, H cf ). Then we repeat the argument leading to (4.3) in [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF]. We reformulate the assumption of ellipticity (2.2.10) as:

dX t = b a (X t )dt + a( σ(X t )dW 1 t + σ0 dW 2 t ),
where σ ≥ 0 and where (W 1 t ) and (W 2 t ) are two independent Brownian motions in R d (which can be expressed in terms of W ). For x ̸ = y, let X x be the solution of this SDE with X 0 = x and let Y y solve the following coupled SDE for Y y 0 = y :

dY y t = b a (Y y t )dt + a σ(Y y t )dW 1 t + a σ0 dW 2 t -2 ⟨X x t -Y y t , dW 2 t ⟩(X x t -Y y t ) |X x t -Y y t | 2 .
The process Y y is in fact defined by orthogonally symmetrizing the component of the noise in W 2 w.r.t. X x t -Y y t at every instant t. This SDE has a unique solution up to the coupling time

T x,y := inf{t ≥ 0 : X x t = Y y t },
and for t ≥ T x,y we set Y y t = X x t . Then Y y has the same distribution as X y i.e. is a weak solution of (2.4.1) with starting value y and it follows from (2.4.5) and from the Itō formula applied to |X x -Y y | that for every 0 ≤ u ≤ t ≤ T x,y ,

|X x t -Y y t | -|X x u -Y y u | ≤ M t -M u + t u (K 1 +K 2 )1 |X x s -Y y s |≤r 0 -K 2 |X x s -Y y s |ds,
where

M t = t 0 a⟨2 σ0 dW 2 s + ( σ(X s ) -σ(Y y s ))dW 1 t , X x s -Y y s ⟩ |X x s -Y y s | is a true Brownian martingale with bracket process satisfying ⟨M ⟩ t ≥ 4a 2 σ2 0 t. (2.4.6)
We now set, still like in the proof of (4.3) in [START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF],

p t := |X x t -Y y t | and pt := εp t + 1 -e -N pt ,
where

N := r 0 a 2 σ2 0 (K 1 + K 2 ) and ε := N e -N r 0 .
Then we have :

εp t ≤ pt ≤ (N + ε)p t , and ∀r ∈ [0, r 0 ), 2N 2 r(εe N r + N ) ≥ K 1 + K 2 a 2 σ2 0 .
Then using (2.4.6) we derive for all 0 ≤ u ≤ t ≤ T x,y :

pt -pu ≤ t u (ε + N e -N ps )dM s + t u (ε + N e -N ps ) (K 1 + K 2 )1 ps≤r 0 -K 2 - 2N 2 a 2 σ2 0 p s (εe N ps + N ) p s ds ≤ Mt -Mu -K 2 t u (ε + N e -N ps )p s ds ≤ Mt -Mu -εK 2 t u p s ds ≤ Mt -Mu - εK 2 N + ε t u ps ds.
So that we have

E[p t -pu ] = E[(p t -pu )1 t≤Tx,y ] ≤ - εK 2 N + ε t u Ep s ds, so that d dt E[p t ] ≤ - εK 2 N + ε E[p t ]
and then

Ep t ≤ p0 e -εK 2 N +ε t .
Noting that p0 ≤ (N + ε)|x -y|, we have

Ep t ≤ N + ε ε |x -y|e -εK 2 N +ε t , .
so that

W 1 ([X x t ] , [X y t ]) ≤ N + ε ε |x -y|e -εK 2 N +ε t ≤ Ce C 1 /a 2 |x -y|e -e -C 2 /a 2 t .
(b) As ν a is the invariant distribution of the diffusion (2.4.1), using (2.4.3, P cf ) we have

W 1 ([X x t ] , ν a ) = R d W 1 ([X x t ] , [X y t ]) ν a (dy) ≤ Ce C 1 /a 2 e -ρat R d |x -y|ν a (dy) ≤ Ce C 1 /a 2 e -ρat W 1 (δ x , ν a ).

Time schedule and Wasserstein distance between Gibbs measures

For C (T ) > 0 and for β > 0, let us define the time schedule that will be used for the plateau SDE in the next section:

T n := C (T ) n 1+β , (2.4.7)
and by a slight abuse of notation we define

a n := a(T n ) = A log(T n + e)
and ρ n := ρ an = e -C 2 /a 2 n .

(2.4.8)

Lemma 2.4.3. The sequence

a n = A log -1/2 (T n + e) satisfies 0 ≤ a n -a n+1 ≍ (n log 3/2 (n)) -1 .
(2.4.9)

Proof. One straightforwardly checks that

a n -a n+1 ∼ - d dn   A log(C (T ) n 1+β + e)   = A(1 + β) 2 log 3/2 (C (T ) n 1+β +e) n + e/(C (T ) n β ) ≍ 1 n log 3/2 (n) .
We prove the following result that will be useful to study the convergence of the plateau SDE.

Proposition 2.4.4. Let ν a , a ∈ (0, A] be the Gibbs measure defined in (2.2.3). Assume that V is coercive, that

(x → |x| 2 e -2V (x)/A 2 ) ∈ L 1 (R d ) and (2.2.2, H V 1 ). Then for n ∈ N, W 1 (ν an , ν a n+1 ) ≤ C n log 3/2 (n) .
Moreover, for every s, t ∈ [a n+1 , a n ], we have

W 1 (ν s , ν t ) ≤ C n log 3/2 (n) .
The proof of this proposition is given in the Supplementary Material. It relies on the following lemma.

Lemma 2.4.5. Let µ and ν be two probability distributions on R d with densities f and g respectively with finite moments of order p. Assume that there exists M ≥ 1 such that f ≤ M g.

Then W p (µ, ν) p ≤ E|X -Y | p - 1 M E|X -X| p ,
where X and X ∼ µ, Y ∼ ν and X, X and Y are mutually independent.

Plateau case

Proof. We define a coupling on µ and ν inspired from the acceptance rejection sampling as follows. Let X ∼ µ, Y ∼ ν, U ∼ U([0, 1]) and X, Y , U are independent, and let

X ′ = Y 1{U ≤ f (Y )/(M g(Y ))} + X1{U > f (Y )/(M g(Y ))}.
Then adapting the proof of the acceptance rejection method, X ′ ∼ µ and we have:

E|X ′ -Y | p = E|Y -X| p 1{U > f (Y )/(M g(Y ))} = (R d ) 2 |y -x| p 1 0 1{u > f (y)/(M g(y))}du f (x)g(y)dxdy = (R d ) 2 |y -x| p f (x)g(y)dxdy - 1 M (R d ) 2 |y -x| p f (x)f (y)dxdy = E|X -Y | p - 1 M E|X -X| p .
Lemma 2.4.6. We have

W 1 (ν an , ν ⋆ ) ≤ Ca n .
(2.4.10)

Proof. First let us prove that W 1 (ν a , ν ⋆ ) → 0 as a → 0. By Proposition 2.4.4 and using that

n≥2 (n log 3/2 (n)) -1 < ∞,
(ν an ) is a Cauchy sequence in (L 1 (R d ), W 1 ) so converges to some limit measure ν. But (ν an ) also weakly converges to ν, so ν = ν ⋆ . Moreover, W 1 (ν an , ν ⋆ ) is bounded by the tail of the above series, which is of order log -1/2 (n).

Remark 2.4.7. Considering ν a (dx) = Z a exp(-2|x| 2 /a 2 )dx with

V (x) = |x| 2 gives W 1 (ν a , ν ⋆ ) = E|Z a -0| = (a/2)E|N (0, I d )|
with Z a ∼ ν a , showing that we cannot get better convergence rates in Lemma 2.4.6 in general.

Plateau case

We define (X t ) as the solution the following SDE where the coefficients piecewisely depend on the time; X is then said to be "by plateaux":

X x 0 0 = x 0 , dX x 0 t = b a k+1 (X x 0 t )dt + a k+1 σ(X x 0 t )dW t , t ∈ [T k , T k+1 ], (2.5.1)
where b a is defined in (2.2.6), (T n ) is defined in (2.4.7) and (a n ) is defined in (2.4.8). We first prove the convergence of (X t ) using ergodic properties of the corresponding SDE on each plateau [T n , T n+1 ]. We note that although the coefficients are not continuous, the process (X x 0 t ) is well defined as it is the continuous concatenation of the solutions of the equations on the intervals [T k , T k+1 ]. More generally, we define (X x,n t ) as the solution of

X x,n 0 = x, dX x,n t = b a k+1 (X x,n t )dt + a k+1 σ(X x,n t )dW t , t ∈ [T k -T n , T k+1 -T n ], k ≥ n,
i.e. (X x,n t ) has the law of (X Tn+t ) t≥0 conditionally to X Tn = x. We have X x t = X x,0 t .

Theorem 2.5.1. Let X be defined in (2.5.1). If

A > max (1 + β -1 )C 2 , (1 + β)C 1 , (2.5.2)
where C 1 and C 2 are given in (2.4.3, P cf ), then for every x 0 ∈ R d :

W 1 ([X x 0 t ], ν ⋆ ) -→ t→∞ 0.
More precisely, for t ≥ 0 we have:

(i) W 1 ([X x 0 t ], ν ⋆ ) ≤ Ca(t)(1 + |x 0 |), (ii) for all C ′ < C (T ) , for all large enough n ≥ n(C ′ (T ) ), W 1 ([X x 0 Tn ], ν an ) ≤ Cn -1+(β+1)C 1 /A 2 e -(C ′ ) 1-C 2 /A 2 (β+1)n β-(β+1)C 2 /A 2 (1 + |x 0 |), (iii) W 1 ([X x 0 t ], ν a(t) ) ≤ C(1 + |x 0 |) t (β+1) -1 -C 1 /A 2 log 3/2 (t) .
Proof. For fixed x ∈ R d and using Theorem 2.4.2 we have:

W 1 ([X x,n T n+1 -Tn ], ν a n+1 ) ≤ Ce C 1 /a 2 n+1 e -ρa n+1 (T n+1 -Tn) W 1 (δ x , ν a n+1 ).
So integrating x with respect to the law of X x 0 Tn (and using the existence of the optimal coupling, see for example [Wan12, Proposition 1.3]) yields:

W 1 ([X x 0 T n+1 ], ν a n+1 ) ≤ Ce C 1 /a 2 n+1 e -ρa n+1 (T n+1 -Tn) W 1 ([X x 0 Tn ], ν an ) + W 1 (ν an , ν a n+1 ) . (2.5.3)
Iterating this relation yields

W 1 ([X x 0 T n+1 ], ν a n+1 ) ≤ µ n+1 W 1 (ν an , ν a n+1 ) + µ n+1 µ n W 1 (ν a n-1 , ν an ) + • • • + µ n+1 • • • µ 1 W 1 (ν a 0 , ν a 1 ) + µ n+1 • • • µ 1 W 1 (δ x 0 , ν a 0 ). (2.5.4)
where

µ n := Ce C 1 /a 2 n e -ρa n (Tn-T n-1 ) = C(T n + e) C 1 /A 2 e -(Tn+e) -C 2 /A 2 (Tn-T n-1 ) ≤ C(C (T ) n β+1 + e) C 1 /A 2 e -(C (T ) n β+1 +e) -C 2 /A 2 C (T ) (β+1)(n-1) β ≤ Cn (β+1)C 1 /A 2 e -(C ′ ) 1-C 2 /A 2 (β+1)n β-(β+1)C 2 /A 2 , (2.5.5)
where we have used (2.4.7) and where the last inequality holds for large enough n. Note that µ n is bounded by a sequence in the form of n δ exp(-Ln η ) = o(n -ℓ ) for every ℓ ≥ 0. Owing to (2.5.2), we have β -

(β + 1)C 2 /A 2 > 0. On the other hand, if Z ∼ ν a 0 then W 1 (δ x 0 , ν a 0 ) = E|x 0 -Z| ≤ |x 0 | + E|Z|.
Plugging this into (2.5.4) and using that µ n → 0 so is bounded and smaller than 1 for n large enough and then (µ n-1 • • • µ k ) 1≤k≤n-1 is bounded ; using Proposition 2.4.4 and that n (n log

3/2 (n)) -1 < ∞ yields W 1 ([X x 0 T n+1 ], ν a n+1 ) ≤ µ n+1 W 1 (ν an , ν a n+1 ) + Cµ n+1 µ n W 1 (ν a n-1 , ν an ) + • • • + W 1 (ν a 0 , ν a 1 ) + Cµ n+1 µ n W 1 (δ x 0 , ν a 0 ) ≤ µ n+1 W 1 (ν an , ν a n+1 ) + Cµ n+1 µ n + Cµ n+1 µ n (1 + |x 0 |) ≤ C µ n+1 n log 3/2 (n) (1 + |x 0 |) ≤ Cµ n+1 a n+1 (1 + |x 0 |),
where we used that µ n = o(W 1 (ν an , ν a n+1 )). Then using Lemma 2.4.6 we have

W 1 ([X x 0 T n+1 ], ν ⋆ ) ≤ W 1 ([X x 0 T n+1 ], ν a n+1 ) + W 1 (ν a n+1 , ν ⋆ ) ≤ Ca n (1 + |x 0 |
), where we used once again µ n → 0. Now, let us prove that W 1 ([X x 0 t ], ν ⋆ ) → 0 as t → ∞. For t ∈ [0, T n+1 -T n ) we integrate (2.4.4) with respect to the law of X x 0 Tn , giving

W 1 ([X x 0 Tn+t ], ν a n+1 ) ≤ Ce C 1 a -2 n+1 e -ρa n+1 t W 1 ([X x 0 Tn ], ν a n+1 ) ≤ Ce C 1 a -2 n+1 W 1 ([X x 0 Tn ], ν an ) + W 1 (ν an , ν a n+1 ) ≤ Ce C 1 a -2 n+1 W 1 (ν an , ν a n+1 )(1 + |x 0 |) ≤ C(1 + |x 0 |) n 1-(β+1)C 1 /A 2 log 3/2 (n) .
(2.5.6)

Now, for t ≥ 0, let n be such that t ∈ [T n , T n+1 ). Then (n + 1) ≥ t 1/(β+1) and

W 1 ([X x 0 t ], ν a(t) ) ≤ W 1 ([X x 0 t ], ν a n+1 ) + W 1 (ν a n+1 , ν a(t) ) ≤ C(1 + |x 0 |) t (β+1) -1 -C 1 /A 2 log 3/2 (t) , (2.5.7)
where we used the second claim of Proposition 2.4.4. Furthermore owing to (2.5.2) we have (β + 1)C 1 /A 2 < 1, so that

W 1 ([X x 0 Tn+t ], ν ⋆ ) ≤ W 1 ([X x 0 Tn+t ], ν a n+1 ) + W 1 (ν a n+1 , ν ⋆ ) ≤ Ca n (1 + |x 0 |).
Remark 2.5.2. We find again the classic schedule a(t) of order log -1/2 (t). If for example we choose instead a n = log(T n ) -(1+ε)/2 for some ε > 0, then we obtain

log(µ 1 • • • µ n ) = n log(C) + C 1 A 2 n k=1 log 1+ε (T k ) - n k=1 T k -T k-1 T log ε (T k )C 2 /A 2 k . Hence, as T n -T n-1 = o(T log ε (Tn)C 2 /A 2 n
), µ 1 • • • µ n does not converge to 0 whatever the value of A > 0 is.

Continuously decreasing case

We now consider (Y t ) solution to (2.2.5) i.e. the Langevin equation where the time coefficient a(t) before σ is continuously decreasing. More generally, since Y is solution to a non-homogeneous SDE, we define for every x ∈ R d and for every fixed u ≥ 0: such that for all f :

Y x 0,u = x, dY x t,u = b a(t+u) (Y x t,u )dt + a(t + u)σ(Y x t,
R d → R + measurable, P Y t,u f (x) = E[f (Y x t,u )].
We also consider X as defined in (2.5.1) and its associated kernel denoted as P X,n t such that for every f :

R d → R + measurable, P X,n t f (x) = E[f (X x,n t )].
In this section we prove the convergence of (Y t ) by giving bounds on W 1 (Y x t , X x t ) and using Theorem 2.5.1.

Boundedness of the potential

Lemma 2.6.1. Let p > 0. Then there exists C > 0 such that for every n ≥ 0, for every u ≥ 0 and for every x ∈ R d :

sup t≥0 EV p (X x,n t ) ≤ CV p (x) and sup t≥0 EV p (Y x t,u ) ≤ CV p (x).
Proof. By the Itō Lemma, we have for k ≥ n and for t ∈ [T k -T n , T k+1 -T n ):

dV p (X x,n t ) = p∇V (X x,n t ) ⊤ • V p-1 (X x,n t )a k+1 σ(X x,n t )dW t + p∇V (X x,n t ) ⊤ • V p-1 (X x,n t ) -σσ ⊤ (X x,n t )∇V (X x,n t ) + a 2 k+1 Υ(X x,n t ) dt + p 2 ∇ 2 V (X x,n t )V p-1 (X x,n t ) + (p -1)|∇V (X x,n t )| 2 • V p-2 (X x,n t ) a 2 k+1 σσ ⊤ (X x,n t )dt.
Using the facts that (a k ), Υ, σ, ∇ 2 V are bounded, that |∇V | ≤ CV 1/2 and that V , |∇V | are coercive and σσ ⊤ ≥ σ2 0 I d , there exists R > 0 such that if |X x,n t | ≥ R then the coefficient of dt in the last equation is bounded above by

pV p-1 ∇V (X x,n t ) T • -σ2 0 (X x,n t )∇V (X x,n t ) + C + CV p-1 (X x,n t )||σ|| 2 ∞ ≤ 0, so that E[V p (X x,n t )] ≤ max sup |z|≤R V p (z), V p (x) .
The proof is the same for Y , replacing a k+1 by a(t).

Strong and weak error bounds

In this subsection we adapt the proofs to bound weak and strong errors from [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] while paying attention to the dependence in a n .

Lemma 2.6.2. Let p ≥ 1 and let γ > 0. There exists C > 0 such that for all n ≥ 0, u, t ≥ 0

such that u ∈ [T n , T n+1 ], u + t ∈ [T n , T n+1 ] and t ≤ γ, ||X x,n t -Y x t,u || p ≤ C √ t(a n -a n+1 ).
Proof. We first consider the case p ≥ 2. Noting that a n+1 ≤ a(u + s) ≤ a n for all s ∈ [0, t] and using Lemma 2.11.1 in the Appendix, with in mind that b a = b 0 + a 2 Υ, we have

∥X x,n t -Y x t,u ∥ p ≤ t 0 (b a n+1 (X x,n s ) -b a(u+s) (Y x s,u ))ds p + t 0 (a n+1 σ(X x,n s ) -a(u + s)σ(Y x s,u ))dW s p ≤ [b] Lip t 0 ||X x,n s -Y x s,u || p ds + t 0 ||a 2 n+1 Υ(X x,n s ) -a(u + s) 2 Υ(Y x s,u )|| p ds + C BDG p a n+1 [σ] Lip t 0 ||X x,n s -Y x s,u || 2 p ds 1/2 + t 0 σ(Y x s,u )(a n+1 -a(u+s))dW s p ≤ [b] Lip t 0 ||X x,n s -Y x s,u || p ds + ||Υ|| ∞ (a 2 n -a 2 n+1 )t + a 2 n+1 [Υ] Lip t 0 ||X x,n s -Y x s,u || p ds + C BDG p a n+1 [σ] Lip t 0 ||X x,n s -Y x s,u || 2 p ds 1/2 + ||σ|| ∞ ||W 1 || p √ t(a n -a n+1 ),
where we used the generalized Minkowski inequality. Set φ(t)

:= sup 0≤s≤t ||X x,n s -Y x s,u || p and ψ(t) := ||Υ|| ∞ (a 2 n -a 2 n+1 )t + ||σ|| ∞ ||W 1 || p √ t(a n -a n+1
). Both functions are non-decreasing and

φ(t) ≤ ψ(t) + ([b] Lip + a 2 n+1 [Υ] Lip ) t 0 ||X x,n s -Y x s,u || p ds + C BDG p a n+1 [σ] Lip t 0 ||X x,n s -Y x s,u || 2 p ds 1/2 .
Moreover, for every η > 0:

t 0 φ(s) 2 ds 1/2 ≤ φ(t) t 0 φ(s)ds ≤ η 2 φ(t) + 1 2η t 0 φ(s)ds. Taking η = C BDG p a n+1 [σ] Lip -1
yields:

φ(t) ≤ 2ψ(t) + 2[b] Lip + 2a 2 n+1 [Υ] Lip + (C BDG a n+1 [σ] Lip ) 2 t 0 φ(s)ds.
So the Gronwall Lemma yields for every t ∈ [0, γ]

φ(t) ≤ 2e (2[b] Lip +2a 2 n+1 [Υ] Lip +(C BDG p a n+1 [σ] Lip ) 2 )γ ψ(t), which completes the proof for p ≥ 2, noting that a 2 n -a 2 n+1 ≤ 2a n (a n -a n+1 ) = o(a n -a n+1 ). If p ∈ [1, 2), the inequality is still true remarking that ∥ • ∥ p ≤ ∥ • ∥ 2 .
Lemma 2.6.3. Let p ≥ 1 and let γ > 0. There exists a real constant C ≥ 0 such that for all

n ≥ 0, ∀t ∈ [0, γ], ||X x,n t -x|| p ≤ CV 1/2 (x) √ t.
Proof. We perform a proof similar to the proof of Lemma 2.6.2. For p ≥ 2 we have

||X x,n t -x|| p ≤ t 0 b a n+1 (X x,n s )ds p + t 0 a n+1 σ(X x,n s )ds p ≤ t|b a n+1 (x)| + A∥σ∥ ∞ ∥W ∥ 1 √ t + [b] Lip t 0 ||X x,n s -x|| p ds + A[σ] Lip C BDG p t 0 ||X x,n s -x|| 2 p ds 1/2
.

From here we use the Gronwall Lemma as in the proof of Lemma 2.6.2. For p ∈ [1, 2), we have

∥ • ∥ p ≤ ∥ • ∥ 2 .
Proposition 2.6.4. Let γ > 0. There exists C > 0 such that for every g : R d → R being C 2 , for every γ ∈ (0, γ], every n ≥ 0 and every

u ≥ 0 such that u ∈ [T n , T n+1 ] and u + γ ∈ [T n , T n+1 ]: |E g(Y x γ,u ) -E g(X x,n γ ) | ≤ Cγ(a n -a n+1 )Φ g (x) with Φ g (x) = max   |∇g(x)|, sup ξ∈(X x,n γ ,Y x γ,u ) ||∇ 2 g(ξ)|| 2 , V 1/2 (x) sup ξ∈(x,X x,n γ ) ||∇ 2 g(ξ)|| 2   .
Proof. By the second order Taylor formula, for every y, z ∈ R d :

g(z) -g(y) = ⟨∇g(y)|z -y⟩ + 1 0 (1 -s)∇ 2 g(sz + (1 -s)y)ds(z -y) ⊗2 .
Applying this expansion with y = X x,n γ and z = Y x γ,u yields:

E[g(Y x γ,u ) -g(X x,n γ )] = ⟨∇g(x)|E[Y x γ,u -X x,n γ ]⟩ + E[⟨∇g(X x,n γ ) -∇g(x), Y x γ,u -X x,n γ ⟩] + 1 0 (1 -s)E ∇ 2 g(sY x γ,u + (1 -s)X x,n γ )(Y x γ,u -X x,n γ ) ⊗2 ds. (2.6.2)
The first term is bounded by

|∇g(x)| • |E[Y x γ,u -X x,n γ ]|, with |E[Y x γ,u -X x,n γ ]| = E γ 0 (b a(s+u) (Y x s,u ) -b a(s+u) (X x,n s ))ds + E γ 0 (b a(u+s) (X x,n s ) -b a n+1 (X x,n s ))ds ≤ C[b] Lip (a n -a n+1 ) γ 0 √ sds + ||Υ|| ∞ γ(a 2 n -a 2 n+1 ) ≤ Cγ(a n -a n+1 ),
where we used Lemma 2.6.2. Using Lemma 2.6.3 and Lemma 2.6.2 again, the second term in the right hand side of (2.6.2) is bounded by

C sup ξ∈(x,X x,n γ ) ||∇ 2 g(ξ)|| 2 √ γV 1/2 (x) √ γ(a n -a n+1 ).
Using Lemma 2.6.2, the third term is bounded by

1 2 Cγ(a n -a n+1 ) 2 sup ξ∈(X x,n γ ,Y x γ,u ) ||∇ 2 g(ξ)|| 2 .
Proposition 2.6.5. Let T , γ > 0. There exists C > 0 such that for every Lipschitz continuous function f : R d → R and every t ∈ (0, T ], for all n ≥ 0, for all γ < γ and every u

∈ [T n , T n+1 ] such that u + t + γ ∈ [T n , T n+1 ], E P X,n t f (Y x γ,u ) -E P X,n t f (X x,n γ ) ≤ Ca -2 n+1 (a n -a n+1 )[f ] Lip γt -1/2 V (x).
Proof. We apply Proposition 2.6.4 to g t := P X,n t f with t > 0. Following [PP23, Proposition 3.2(b)] while paying attention to the dependence in the ellipticity parameter a, we have

Φ gt (x) ≤ C[f ] Lip a -2 n+1 t -1/2 max   V 1/2 (x), sup ξ∈(X x,n γ ,Y x γ,u ) V 1/2 (ξ) 2 , V 1/2 (x) sup ξ∈(x,X x,n γ ) V 1/2 (ξ) 2   . But following (2.2.8, H V 2 ), ∇V /V 1/2 is bounded so V 1/2 is Lipschitz continuous and then sup ξ∈(x,X x,n γ ) V 1/2 (ξ) 2 ≤ V 1/2 (x) + [V 1/2 ] Lip |X x,n γ -x| 2 ≤ CV 1/2 (x) sup ξ∈(X x,n γ ,Y x γ,u ) V 1/2 (ξ) 2 ≤ V 1/2 (x) + [V 1/2 ] Lip max(|X x,n γ -x|, |Y x γ,u -x|) 2 ≤ CV 1/2 (x),
where we used Lemmas 2.6.3 and 2.6.2. We thus obtain the desired result.

T n T n+1

T n+1 -T γ Figure 2.1: Intervals for the domino strategy.

Proof of Theorem 2.2.1.(a)

More precisely, we prove that for all β > 0, if

A > max (β + 1)(2C 1 + C 2 ), (1 + β -1 )C 2 , (2.6.3) then W 1 ([Y x 0 t ], ν a(t) ) ≤ C max(1 + |x 0 |, V (x 0 )) log 3/2 (t)t (1+β) -1 -(2C 1 +C 2 )/A 2 .
Proof. We apply the domino strategy (2.1.4). Let us fix T ∈ (0, T 1 ) and γ ∈ (0, T 1 -T ). Here γ is not linked to any Euler-Maruyama scheme but is an auxiliary tool for the proof. Let n ≥ 0 and let f : R d → R be Lipschitz continuous. We divide the two intervals [T n , T n+1 -T ] and [T n+1 -T, T n+1 ] into smaller intervals of size γ (see Figure 2.1) and for x ∈ R d using the semi-group property of P X,n on [T n , T n+1 ) we write:

Ef (X x,n T n+1 -Tn ) -Ef (Y x T n+1 -Tn,Tn ) ≤ ⌊(T n+1 -Tn-T )/γ⌋ k=1 P Y (k-1)γ,Tn • (P Y γ,Tn+(k-1)γ -P X,n γ ) • P X,n T n+1 -Tn-kγ f (x) + ⌊(T n+1 -Tn)/γ⌋-1 k=⌊(T n+1 -Tn-T )/γ⌋+1 P Y (k-1)γ,Tn • (P Y γ,Tn+(k-1)γ -P X,n γ ) • P X,n T n+1 -Tn-kγ f (x)
+ P Y γ(⌊(T n+1 -Tn)/γ⌋-1),Tn • (P Y γ+(T n+1 -Tn)modγ,Tn+γ(⌊(T n+1 -Tn)/γ⌋-1) -P X,n γ+(T n+1 -Tn)mod(γ) )f (x) =: (a) + (b) + (c).

The term (a) is the "ergodic term", for which the exponential contraction from Theorem 2.4.2 can be exploited. The terms (b) and (c) are the "error terms" where we bound the error on intervals of length no larger than T . The term (c) is a remainder term due to the fact that T n+1 -T n is generally not a multiple of γ.

• Term (a) : It follows from Theorem 2.4.2 and Lemma 2.6.2 that

|(P Y γ,Tn+(k-1)γ -P X,n γ ) • P X,n T n+1 -Tn-kγ f (x)| = |EP X,n T n+1 -Tn-kγ f (X x,n γ ) -EP X T n+1 -Tn-kγ,n f (Y x γ,Tn+(k-1)γ )| ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn-kγ) [f ] Lip E|X x,n γ -Y x γ,Tn+(k-1)γ | ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn-kγ) [f ] Lip √ γ(a n -a n+1 )
Integrating with respect to P Y (k-1)γ,Tn and summing up yields

(a) ≤ Ce C 1 a -2 n+1 [f ] Lip √ γ(a n -a n+1 ) e -ρ n+1 T -e -ρ n+1 (T n+1 -Tn) e γρ n+1 -1 ≤ Ce C 1 a -2 n+1 [f ] Lip √ γ(a n -a n+1 )(γρ n+1 ) -1 .
• Term (b): Applying Proposition 2.6.5 yields:

|(P Y γ,Tn+(k-1)γ -P X,n γ ) • P X,n T n+1 -Tn-kγ f (x)| ≤ Ca -2 n+1 (a n -a n+1 )[f ] Lip γ T n+1 -T n -kγ V (x).
Integrating with respect to P Y (k-1)γ,Tn and using Lemma 2.6.1 which guarantees that P Y (k-1)γ,Tn V (x) ≤ CV (x) and summing with respect to k implies

(b) ≤ Ca -2 n (a n -a n+1 )[f ] Lip γV (x) ⌈T /γ⌉ k=1 (kγ) -1/2 ≤ Ca -2 n (a n -a n+1 )[f ] Lip T 1/2 V (x).
• Term (c):

Noting that γ + (T n+1 -T n ) mod(γ) ≤ 2γ, Lemma 2.6.2 yields (c) ≤ C[f ] Lip √ γ(a n -a n+1 ).
Now we sum up the terms (a), (b) and (c). Since γ is constant we have:

Ef (X x,n T n+1 -Tn ) -Ef (Y x T n+1 -Tn,Tn ) ≤ Ce C 1 a -2 n+1 (a n -a n+1 )ρ -1 n+1 [f ] Lip V (x), so that for all x ∈ R d , W 1 ([X x,n T n+1 -Tn ], [Y x T n+1 -Tn,Tn ]) ≤ Ce C 1 a -2 n+1 (a n -a n+1 )ρ -1 n+1 V (x).
(2.6.4)

Temporarily setting x n := X x 0 Tn and y n := Y x 0 Tn , we derive

W 1 ([X x 0 T n+1 ], [Y x 0 T n+1 ]) = W 1 ([X xn,n T n+1 -Tn ], [Y yn T n+1 -Tn,Tn ]) ≤ W 1 ([X xn,n T n+1 -Tn ], [X yn,n T n+1 -Tn ]) + W 1 ([X yn,n T n+1 -Tn ], [Y yn T n+1 -Tn,Tn ]) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn) W 1 ([X x 0 Tn ], [Y x 0 Tn ]) + Ce C 1 a -2 n+1 (a n -a n+1 )ρ -1 n+1 EV (Y x 0 Tn ),
where we used Theorem 2.4.2 and (2.6.4). We then apply Lemma 2.6.1 which guarantees that (EV (Y x 0 Tn )) n is bounded by CV (x 0 ). Let us denote

λ n := Ce C 1 a -2 n (a n-1 -a n )ρ -1 n = Ce (C 1 +C 2 )a -2 n (a n-1 -a n ).
Owing to (2.6.3) we have λ n → 0. Iterating this relation and using (µ n ) defined in (2.5.5) yields like in the proof of Theorem 2.5.1:

W 1 ([X x 0 T n+1 ], [Y x 0 T n+1 ]) ≤ CV (x 0 ) (λ n+1 + µ n+1 λ n + µ n+1 µ n λ n-1 + • • • + µ n+1 • • • µ 2 λ 1 ) ≤ CV (x 0 ) (λ n+1 + µ n+1 (λ n + • • • + λ 1 )) ≤ CV (x 0 ) (λ n+1 + nµ n+1 ) .
But following (2.5.5) one checks that nµ n+1 = o(λ n+1 ) so that

W 1 ([X x 0 T n+1 ], [Y x 0 T n+1 ]) ≤ CV (x 0 )λ n+1 ≤ CV (x 0 ) log 3/2 (n + 1)(n + 1) 1-(β+1)(C 1 +C 2 )/A 2 .
Moreover, owing to (2.6.3) and combining with Theorem 2.5.1 we get

W 1 ([Y x 0 Tn ], ν an ) ≤ W 1 ([Y x 0 Tn ], [X x 0 Tn ]) + W 1 ([X x 0 Tn ], ν an ) ≤ C max(1 + |x 0 |, V (x 0 )) log 3/2 (n)n 1-(β+1)(C 1 +C 2 )/A 2 T 1 T 2 T 3 T 4 t a(t)
Non plateau case Plateau case Figure 2.2: Decreasing of the noise coefficient a for the plateau and non plateau cases.

and as the right hand side of these inequalities is in o(a n ), we derive

W 1 ([Y x 0 Tn ], ν ⋆ ) ≤ W 1 ([Y x 0 Tn ], [X x 0 Tn ]) + W 1 ([X x 0 Tn ], ν ⋆ ) ≤ Ca n max(1 + |x 0 |, V (x 0 )). • Convergence for t → ∞ : Now let us prove that W 1 ([Y x 0 t ], ν ⋆ ) → 0 as t → ∞.
As before, let T > 0. For t ≥ T , then we perform the same domino strategy where we replace T n+1 by T n + t and we consider the intervals [T n , T n + t -T ] and [T n + t -T, T n + t]. For t < T then we only consider the terms (b) and (c) and we replace T by t in (b). Doing so we obtain

W 1 ([X x,n t ], [Y x t,Tn ]) ≤ Ce C 1 a -2 n+1 (a n -a n+1 )ρ -1 n+1 V (x)
. So that, as before:

W 1 ([X x 0 Tn+t ], [Y x 0 Tn+t ]) ≤ Ce C 1 a -2 n+1 W 1 ([X x 0 Tn ], [Y x 0 Tn ]) + Ce C 1 a -2 n+1 (a n -a n+1 )ρ -1 n+1 V (x 0 ) ≤ CV (x 0 ) log 3/2 (n)n 1-(β+1)(2C 1 +C 2 )/A 2 .
Owing to (2.6.3) we have 1 -(β + 1)(2C 1 + C 2 )/A 2 > 1, so that, using (2.5.6),

W 1 ([Y x 0 Tn+t ], ν a n+1 ) ≤ W 1 ([Y x 0 Tn+t ], [X x 0 Tn+t ]) + W 1 ([X x 0 Tn+t ], ν a n+1 ) ≤ C max(1 + |x 0 |, V (x 0 )) log 3/2 (n)n 1-(β+1)(2C 1 +C 2 )/A 2 .
We then prove the bound for W 1 ([Y x 0 t ], ν a(t) ) the same way as for (2.5.7), using the second claim of Proposition 2.4.4.

Continuously decreasing case : the Euler-Maruyama scheme

We now consider ( Ȳn ) to be the Euler-Maruyama scheme of (Y t ) with steps (γ n ) defined in (2.2.15) and we also consider its genuine interpolation defined in (2.2.16). In this section we prove the convergence of ( Ȳt ) by giving bounds on W 1 (Y x t , X x t ) and using Theorem 2.5.1. As with (2.6.1), we define more generally for every n ≥ 0, (

Ȳ x t,Γn ) t≥0 , first at times Γ k -Γ n , k ≥ n, by Ȳ x 0,Γn = x, Ȳ x Γ k+1 -Γn,Γn = Ȳ x Γ k -Γn,Γn + γ k+1 b a(Γ k ) ( Ȳ x Γ k -Γn,Γn ) + ζ k+1 ( Ȳ x Γ k -Γn,Γn ) + a(Γ k )σ( Ȳ x Γ k -Γn,Γn )(W Γ k+1 -W Γ k
), then at every time t by the genuine interpolation on the intervals ([Γ k -Γ n , Γ k+1 -Γ n )) k≥n as before. In particular Ȳ x = Ȳ x •,0 . Still more generally, we define Ȳ x t,u where u ∈ (Γ n , Γ n+1 ) as

Ȳ x 0,u = x, Ȳ x t,u =    x + t(b a (x) + ζ n+1 (x)) + a 2 (u)σ(x)(W t -W Γu ) if t ∈ [u, Γ n+1 ] Ȳ Ȳ x Γ n+1 -u,u t-(Γ n+1 -u),Γ n+1 if t > Γ n+1 .
For n, k ≥ 0, for u ∈ [Γ k , Γ k+1 ) and γ ∈ [0, Γ k+1 -u], let P Ȳ γ,u be the transition kernel associated to Ȳ•,u between the times 0 and γ i.e. for all f :

R d → R + measurable, P Ȳ γ,u f (x) = E[f ( Ȳ x γ,u )].

Boundedness of the potential

Lemma 2.7.1. Let p ≥ 1/2. There exists a constant C > 0 such that for every k ≥ 0, for every u ∈ [Γ k , Γ k+1 ) and for every x ∈ R d :

sup n≥k+1 EV p ( Ȳ x Γn-u,u ) ≤ CV p (x).
Proof. We rework the proof of Lemma 2(b) in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diffusion[END_REF]. Let us assume directly that u = Γ k .

To simplify the notations, we define y n := Ȳ x Γn-Γ k ,Γ k for n ≥ k and ∆ y n+1 := y n+1 -y n . The Taylor formula applied to V p between y n and y n+1 yields for some ξ n+1 ∈ ( y n , y n+1 ) and with

∇ 2 (V p ) = p(V p-1 ∇ 2 V + (p -1)V p-2 ∇V ∇V T ): V p ( y n+1 ) = V p ( y n ) + pV p-1 ⟨∇V ( y n ), ∆ y n+1 ⟩ + 1 2 ∇ 2 (V p )(ξ n+1 ) • (∆ y n+1 ) ⊗2 = V p ( y n ) + pV p-1 ∇V ( y n ) T • -γ n+1 σσ ⊤ ( y n )∇V ( y n ) + γ n+1 a 2 (Γ n )Υ( y n ) + γ n+1 ζ n+1 ( y n ) + √ γ n+1 a(Γ n )σ( y n )U n+1 + 1 2 ∇ 2 (V p )(ξ n+1 ) • (∆ y n+1 ) ⊗2 ,
where U n+1 ∼ N (0, I d ). Moreover using (2.2.8,

H V 2 ), √ V is Lipschitz continuous so E sup z∈( yn, y n+1 ) V 1/2 (z)| y 1 , . . . , y n ≤ V 1/2 ( y n ) + [ √ V ] Lip E[| y n+1 -y n || y 1 , . . . , y n ] ≤ CV 1/2 ( y n ), (2.7.1)
and in particular

E[∥∇ 2 (V p )(ξ n+1 )∥| y 1 , . . . , y n ] ≤ C∥∇ 2 (V p )( y n )∥. Moreover using that ∇ 2 V is bounded and that |∇V | ≤ CV 1/2 we have ∥∇ 2 (V p )( y n )∥ ≤ C∥(V p-1 ∇ 2 V + V p-2 ∇V ∇V T )( y n )∥ ≤ CV p-1 ( y n ).
Then using the facts that a, Υ, σ, ∇ 2 V are bounded and that γ 2 n = o(γ n ), that V , ∇V are coercive and σσ ⊤ ≥ σ2 0 I d and (2.2.14), there exists R > 0 and

N ∈ N such that if | y n | ≥ R and n ≥ N then E[V p ( y n+1 ) -V p ( y n )| y 1 , . . . , y n ] ≤ pV p-1 ∇V ( y n ) T • -γ n+1 σ2 0 ( y n )∇V ( y n ) + Cγ n+1 + C∥∇ 2 (V p )( y n )∥ • γ 2 n+1 ∥σ∥ 4 ∞ |∇V ( y n )| 2 + Cγ 2 n+1 + Cγ 2 n+1 V (x) + Cγ n+1 E|N (0, I d )| 2 ≤ Cγ n+1 V p-1 ( y n ) |∇V ( y n )| -|∇V ( y n )| + 1 + γ n+1 (|∇V ( y n )| 2 + 1) + 1 ≤ 0. On the other side, if | y n | ≤ R then E[|V p ( y n+1 ) -V p ( y n )|| y 1 , . . . , y n ] ≤ Cγ n+1 sup |x|≤R V p (x).
Moreover for n ∈ {k, . . . , N } using (2.7.1) we have

E[|V p ( y n+1 ) -V p ( y n )|| y 1 , . . . , y n ] ≤ CV p ( y n ) so that sup k≤n≤N +1 E[V p ( y n )] ≤ C N -k V p (x).
Finally we obtain sup n≥k E[V p ( y n )] ≤ CV p (x).

Strong and weak error bounds for the Euler-Maruyama scheme

Lemma 2.7.2. Let p ≥ 1. There exists C > 0 such that for every n, k ≥ 0, for every

u ∈ [Γ k , Γ k+1 ) and every t > 0 such that u ∈ [T n , T n+1 ], t ≤ Γ k+1 -u and u + t ∈ [T n , T n+1 ], ||X x,n t -Ȳ x t,u || p ≤ C V 1/2 (x)t + √ t(a n -a n+1 ) .
Proof. As in the proof of Lemma 2.6.2, if p ≥ 2 we have

||X x,n t -Ȳ x t,u || p ≤ [b] Lip t 0 ||X x,n s -x|| p ds + ||Υ|| ∞ (a 2 n -a 2 n+1 )t + a 2 n+1 [Υ] Lip t 0 ||X x,n s -x|| p ds + ∥ζ 1 (x)∥ p t + C BDG a n+1 [σ] Lip t 0 ||X x,n s -x|| 2 p ds 1/2 + ||σ|| ∞ ||W 1 || p √ t(a n -a n+1 ).
Plugging Lemma 2.6.3 and (2.2.14) into this inequality yields:

||X x,n t -Ȳ x t,u || p ≤ CV 1/2 (x)t 3/2 + ||Υ|| ∞ (a 2 n -a 2 n+1 )t + CV 1/2 (x)t + C √ t(a n -a n+1 ), which completes the proof for p ≥ 2. If p ∈ [1, 2), we remark that ∥ • ∥ p ≤ ∥ • ∥ 2 .
Proposition 2.7.3. For every g :

R d → R being C 3 , for every n, k ≥ 0 and every u ∈ [Γ k , Γ k+1 ) such that u ∈ [T n , T n+1 ], γ ≤ Γ k+1 -u and u + γ ∈ [T n , T n+1 ]: |E g( Ȳ x γ,u ) -E g(X x,n γ ) | ≤ CV 1/2 (x) V 1/2 (x)γ 2 + γ(a n -a n+1 ) Φg,1 (x) + CV (x) V 1/2 (x)γ 2 + γ 3/2 (a n -a n+1 ) Φg,2 (x), with Φg,1 (x) = max   |∇g(x)|, ||∇ 2 g(x)||, sup ξ∈(X x,n γ , Ȳ x γ,u ) ||∇ 2 g(ξ)|| 2   , Φg,2 (x) = sup ξ∈(x,X x,n γ ) ||∇ 3 g(ξ)|| 4 .
The proof is given in the Supplementary Material.

Proposition 2.7.4. Let T > 0. There exists C > 0 such that for every Lipschitz continuous function f and every t ∈ (0, T ], for all n, k ≥ 0, for all u ∈ [Γ k , Γ k+1 ), for all γ such that

Γ k ∈ [T n , T n+1 ], γ ≤ Γ k+1 -u and u + t + γ ∈ [T n , T n+1 ], E P X,n t f ( Ȳ x γ,u ) -E P X,n t f (X x,n γ ) ≤ C[f ] Lip V 2 (x) • a -2 n+1 t -1/2 γ 2 + (a n -a n+1 )γ + a -3 n+1 t -1 γ 2 + γ 3/2 (a n -a n+1 ) .
Proof. The proof is the same as for Proposition 2.6.5.

Proof of Theorem 2.2.1.(b)

More precisely, we prove that for all β > 0, if

A > max (β + 1)(2C 1 + C 2 ), (1 + β -1 )C 2 , (2.7.2) then W 1 ([ Ȳ x 0 t ], ν a(t) ) ≤ C max 1 + |x 0 |, V 2 (x 0 ) t (1+β) -1 -(2C 1 +C 2 )/A 2 .
Proof. We apply the same domino strategy as in Section 2.6.3. Let n ≥ 0 and let f : R d → R be Lipschitz continuous. Let us denote

γ init := Γ N (Tn)+1 -T n ≤ γ N (Tn)+1 and γ end := T n+1 -Γ N (T n+1 ) ≤ γ N (T n+1 )+1 .
For x ∈ R d we write:

Ef (X x,n T n+1 -Tn ) -Ef ( Ȳ x T n+1 -Tn,Tn ) ≤ (P Ȳ γ init ,Tn -P X,n γ init ) • P X,n T n+1 -Γ N (Tn)+1 f (x) + N (T n+1 -T ) k=N (Tn)+2 P Ȳ γ init ,Tn • P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ k-1 ,Γ k-2 • (P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x) + N (T n+1 )-1 k=N (T n+1 -T )+1 P Ȳ γ init ,Tn • P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ k-1 ,Γ k-2 • (P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x) + P Ȳ γ init ,Tn •P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ N (T n+1 )-1 ,Γ N (T n+1 )-2 • (P Ȳ γ end +γ N (T n+1 ) ,Γ N (T n+1 )-1 -P X,n γ end +γ N (T n+1 ) )f (x) =: (c init ) + (a) + (b) + (c end ).
• Term (a): we have

|(P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x)| = |P Ȳ γ k ,Γ k-1 • P X,n T /2 • P X,n T n+1 -Γ k -T /2 f (x) -P X,n γ k ,n • P X,n T /2 • P X,n T n+1 -Γ k -T /2 f (x)| ≤ |EP X,n T n+1 -Γ k -T /2 (Ξ x k ) -EP X,n T n+1 -Γ k -T /2 ( Ξx k )| ≤ Ce Ca -2 n+1 e -ρ n+1 (T n+1 -Γ k -T /2) [f ] Lip E|Ξ x k -Ξx k |,
where Ξ x k and Ξx k are any random vectors with laws X

X x,n γ k ,n T /2 and X Ȳ x γ k ,Γ k-1 ,n T /2
respectively and where we used Theorem 2.4.2 to get the last inequality. Thus, it follows from the definition of the Wasserstein distance that

|(P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x)| ≤ Ce Ca -2 n+1 e -ρ n+1 (T n+1 -Γ k ) [f ] Lip W 1 X X x,n γ k ,n T /2 , X Ȳ x γ k ,Γ k-1 ,n T /2 .
On the other hand, the Kantorovich-Rubinstein representation of the L 1 -Wasserstein distance (see [START_REF] Villani | Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften[END_REF]Equation (6.3)]) reads

W 1 X X x,n γ k ,n T /2 , X Ȳ x γ k ,Γ k-1 ,n T /2 = sup [g] Lip =1 E g X X x,n γ k ,n T /2 -g X Ȳ x γ k ,Γ k-1 ,n T /2 = sup [g] Lip =1 E P X,n T /2 g(X x,n γ k ) -P X,n T /2 g( Ȳ x γ k ,Γ k-1 ) .
It follows from Proposition 2.7.4 and using [g] Lip = 1 that

E P X,n T /2 g(X x,n γ k ) -P X,n T /2 g( Ȳ x γ k ,Γ k-1 ) ≤ Ca -3 n+1 γ 2 k + (a n -a n+1 )γ k V 2 (x),
so that

|(P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x)| ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Γ k ) [f ] Lip a -3 n+1 γ 2 k + (a n -a n+1 )γ k V 2 (x).
Finally, integrating with respect to

P Ȳ γ init ,Tn • P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ k-1 ,Γ k-2 yields: P Ȳ γ init ,Tn • P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ k-1 ,Γ k-2 • (P Ȳ γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Γ k ) [f ] Lip a -3 n+1 γ 2 k + (a n -a n+1 )γ k • sup ℓ≥N (Tn)+1 EV 2 ( Ȳ x γ init +Γ ℓ -Γ N (Tn)+1 ,Tn ) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Γ k ) [f ] Lip a -3 n+1 γ 2 k + (a n -a n+1 )γ k V 2 (x),
where we used Lemma 2.7.1. Now, summing up over k yields:

(a) ≤ Ca -3 n+1 e C 1 a -2 n+1 e -ρ n+1 T n+1 [f ] Lip V 2 (x) N (T n+1 -T ) k=N (Tn)+2 ((a n -a n-1 ) + γ k )γ k e ρ n+1 Γ k ≤ Ca -3 n+1 e C 1 a -2 n+1 e -ρ n+1 T n+1 [f ] Lip ((a n -a n-1 ) + γ N (Tn) )V 2 (x) N (T n+1 -T ) k=N (Tn)+2 γ k e ρ n+1 Γ k-1 ≤ Ca -3 n+1 e C 1 a -2 n+1 e -ρ n+1 T n+1 [f ] Lip ((a n -a n-1 ) + γ N (Tn) )V 2 (x) T n+1 -T Tn e ρ n+1 u du ≤ Ca -3 n+1 e C 1 a -2 n+1 [f ] Lip ((a n -a n-1 ) + γ N (Tn) )V 2 (x)ρ -1 n+1 ≤ Ca -3 n+1 e C 1 a -2 n+1 [f ] Lip (a n -a n-1 )V 2 (x)ρ -1 n+1 ,
where we used that (e ρ n+1 γ k ) n,k≥0 is bounded and Lemma 2.11.3 in the last inequality. We obtain likewise

(c init ) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn) [f ] Lip a -3 n+1 (a n -a n+1 )γ N (Tn)+1 V 2 (x).
• Term (b): Applying Proposition 2.7.4 yields:

(b) ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) • [f ] Lip V 2 (x) N (T n+1 )-1 k=N (T n+1 -T )+1 γ k T n+1 -Γ k + Ca -2 n+1 γ N (T n+1 -T ) + (a n -a n+1 ) [f ] Lip V 2 (x) N (T n+1 )-1 k=N (T n+1 -T )+1 γ k √ T n+1 -Γ k ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) • [f ] Lip V 2 (x) T n+1 -γ N (T n+1 ) T n+1 -T 1 T n+1 -u du + Ca -2 n+1 γ N (T n+1 -T ) + (a n -a n+1 ) [f ] Lip V 2 (x) T n+1 -γ N (T n+1 ) T n+1 -T 1 √ T n+1 -u du ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) [f ] Lip V 2 (x) log(1/γ N (T n+1 ) ) + Ca -2 n+1 (a n -a n+1 )[f ] Lip V 2 (x). Using Lemma 2.11.4 in Appendix, √ γ N (T n+1 -T ) log(1/γ N (T n+1 ) ) ≤ C √ γ N (T n+1 ) log(1/γ N (T n+1 ) ) → 0
and using Lemma 2.11.3 we also have

γ N (T n+1 -T ) log(1/γ N (T n+1 ) ) ≤ Cγ 1-ε N (T n+1 ) = o n -1-β ′ = o(a n -a n+1 )
where

β ′ > 0 for small enough ε. So that (b) ≤ Ca -3 n+1 (a n -a n+1 )[f ] Lip V 2 (x).
• Term (c end ): Using Lemma 2.7.2 and γ end ≤ γ N (T n+1 )+1 ≤ γ N (Tn) yields:

|(P Ȳ γ end +γ N (T n+1 ) ,Γ N (T n+1 )-1 -P X,n γ end +γ N (T n+1 ) )f (x)| ≤ C[f ] Lip √ γ N (Tn) (a n -a n+1 ) + γ N (Tn) V 1/2 (x).
Then we integrate with respect to

P Ȳ γ init ,Tn • P Ȳ γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Ȳ γ k-1 ,Γ k-2
and apply Lemma 2.7.1.

• So we have finally that |Ef (X x,n T n+1 -Tn ) -Ef ( Ȳ x T n+1 -Tn,Tn )| is bounded by

Ca -3 n+1 [f ] Lip (a n -a n+1 )e C 1 a -2 n+1 ρ -1 n+1 V 2 (x), which implies that, for every x ∈ R d , W 1 ([X x,n T n+1 -Tn ], [ Ȳ x T n+1 -Tn,Tn ]) ≤ Ca -3 n+1 (a n -a n+1 )e C 1 a -2 n+1 ρ -1 n+1 V 2 (x).
We integrate this inequality with respect to the laws of X x 0 Tn and Ȳ x 0 Tn and obtain, temporarily setting x n := X x 0 Tn and ȳn := Ȳ x 0 Tn ,

W 1 ([X x 0 T n+1 ], [ Ȳ x 0 T n+1 ]) = W 1 ([X xn,n T n+1 -Tn ], [ Ȳ ȳn T n+1 -Tn,Tn ]) ≤ W 1 ([X xn,n T n+1 -Tn ], [X ȳn,n T n+1 -Tn ]) + W 1 ([X ȳn,n T n+1 -Tn ], [ Ȳ ȳn T n+1 -Tn,Tn ]) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn) W 1 ([X x 0 Tn ], [ Ȳ x 0 Tn ]) + Ca -3 n+1 (a n -a n+1 )e C 1 a -2 n+1 ρ -1 n+1 EV 2 ( Ȳ x 0 Tn ) ≤ Ce C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn) W 1 ([X x 0 Tn ], [ Ȳ x 0 Tn ]) + Ca -3 n+1 (a n -a n+1 )e C 1 a -2 n+1 ρ -1 n+1 V 2 (x 0 ) =: µ n+1 W 1 ([X x 0 Tn ], [ Ȳ x 0 Tn ]) + v n+1 V 2 (x 0 )
, where µ n is defined in (2.5.5) and where we used again Lemma 2.7.1. We use Lemma 2.4.3 to bound (a n -a n+1 ) and owing to (2.7.2) we have v n → 0, so is bounded. We iterate this inequality and obtain

W 1 ([X x 0 T n+1 ], [ Ȳ x 0 T n+1 ]) ≤ CV 2 (x 0 ) (v n+1 + µ n+1 v n + µ n+1 µ n v n-1 + • • • + µ n+1 • • • µ 2 v 1 ) ≤ CV 2 (x 0 ) (v n+1 + Cnµ n+1 ) .
But following (2.5.5) we have

nµ n = O(v n ) so that W 1 ([X x 0 T n+1 ], [ Ȳ x 0 T n+1 ]) ≤ CV 2 (x 0 )v n+1 ≤ CV 2 (x 0 ) (n + 1) 1-(β+1)(C 1 +C 2 )/A 2 .
Moreover, owing to (2.7.2) and combining with Theorem 2.5.1 we get

W 1 ([ Ȳ x 0 Tn ], ν an ) ≤ W 1 ([ Ȳ x 0 Tn ], [X x 0 Tn ]) + W 1 ([X x 0 Tn ], ν an ) ≤ C max(1 + |x 0 |, V 2 (x 0 )) n 1-(β+1)(C 1 +C 2 )/A 2 and W 1 ([ Ȳ x 0 Tn ], ν ⋆ ) ≤ W 1 ([ Ȳ x 0 Tn ], [X x 0 Tn ]) + W 1 ([X x 0 Tn ], ν ⋆ ) ≤ Ca n max(1 + |x 0 |, V 2 (x 0 )).
Finally, to prove that W 1 ([ Ȳ x 0 t ], ν ⋆ ) → 0 as t → ∞, we conclude as in the end of Section 2.6.3.

Convergence of the Euler-Maruyama scheme with plateau

In this section, we consider the Euler-Maruyama scheme for (X t ), that is

Xx 0 0 = x 0 , Xx 0 Γ k+1 = XΓ k + γ k+1 b a n+1 ( XΓ k ) + a n+1 σ( XΓ k )(W Γ k+1 -W Γ k ) for k ∈ {N (T n ), . . . , N (T n+1 ) -1}.
We also define as in Section 2.7 the genuine time-continuous scheme and the Euler-Maruyama scheme for (X x,n t ) t so that Xx 0 ,0 = Xx . Although we already proved the convergence of the Euler-Maruyama scheme for (Y t ), we shall also prove the convergence of the present scheme, since this algorithm is also used by practitioners within the framework of batch methods.

Theorem 2.8.1. Assume (2.2.2, H V 1 ), (2.2.8, H V 2 ), (2.2.9, H σ ), (2.2.10) and (2.2.11, H cf ). Assume furthermore (2.2.12, H γ1 ) and (2.2.13, H γ2 ), that V is C 3 with ∥∇ 3 V ∥ ≤ CV 1/2 and that σ is C 3 with ∥∇ 3 (σσ ⊤ )∥ ≤ CV 1/2 . Then for large enough A > 0 and for every x 0 ∈ R d , W 1 ( Xx 0 t , ν ⋆ ) -→ t→∞ 0.
The proof of this theorem is given in the Supplementary Material.

Experiments

In this section, we compare the performances of adaptive Langevin-Simulated Annealing algorithms versus vanilla SGLD, that is the Langevin algorithm with constant (additive) σ1 . We train an artificial neural network on the MNIST dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], which is composed of grayscale images of size 28 × 28 of handwritten digits (from 0 to 9). The goal is to recognize the handwritten digit and to classify the images. 60000 images are used for training and 10000 images are used for test.

As done by practitioners in general, we do not include the correction term Υ in the Langevin optimizers as computing second-order derivatives highly increases the training time. Moreover

(n) = A log -1/2 (c 1 n + e) with A = 2.10 -3 and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n)
where c 2 n = 1 after 5 epochs and where for SGLD, γ 1 = 0.001 for L-RMSprop and L-Adam and γ 1 = 0.1 for L-Adadelta. the time step coefficient of Υ is a 2 n γ n with a n ≪ 1 and γ n ≪ 1, in comparison with γ n for the gradient and a n γ 1/2 n for the Gaussian noise. We refer to Chapter 5 for more details. We consider a feedforward neural network with two hidden dense layers with 128 units each and with ReLU activation. For the adaptive Langevin algorithms, we choose the function σ as a diagonal matrix which is the square root of the preconditioner in RMSprop [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF], in Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] and in Adadelta [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] respectively (see also Section 2.3.2), giving L-RMSprop, L-Adam and L-Adadelta respectively. The results are given in Figure 2.3 and in Table 2.1.

Preconditioner SGLD L-RMSprop L-Adam L-
As pointed out in the literature (see the references Section 2.3.2), the preconditioned Langevin algorithms show significant improvement compared with the vanilla SGLD algorithm. The convergence is faster and they achieve a lower error on the test set. We also display the value of the loss function on the train set during the training to show that the better performances of the preconditioned algorithms are not due to some overfitting effect.

We also compare preconditioned Langevin algorithms with their respective non-Langevin counterpart. For shallow neural networks, adding an exogenous noise does not seem to improve significantly the performances of the optimization algorithm. However, for deep neural networks, which are highly non-linear and which loss function has many local minima, the Langevin version is competitive with the currently widely used non-Langevin algorithms and can even lead to improvement. The results are given in Figure 2.4 where we used a deep neural network with 20 hidden layers with 32 units each and with ReLU activation.

In order to understand how sensitive are these methods to poor initialization, we run an . We plot the accuracy on the test set in Figure 2.5. We observe that the non-Langevin optimizer needs some time before escaping from the neighbourhood of the initial point whereas in its Langevin version, the Gaussian noise is effective to rapidly escape from highly degenerated saddle points of the loss.

Conclusion and perspectives of future work

We proved the convergence of adaptive Langevin-Simulated annealing algorithms using both ergodic properties and weak error analysis, which fills a gap between the theory, which focused on the additive case, and the practice, where adaptive algorithm are commonly used to obtain faster convergence rates. Dealing with non constant diffusion coefficient σ makes the study more difficult as we need to rely on fine stochastic analysis. We finally obtained the same annealing schedule in log -1/2 (t) as in the additive case. Future work would include relaxation of the assumptions on the coefficients of the SDE, in particular on the growth the potential V at infinity, as it is done for the additive case in [START_REF] Monmarché | Simulated annealing in R d with slowly growing potentials[END_REF]. Another perspective would include weak error analysis of the corresponding projected Langevin-Simulated Annealing algorithm for optimization under constraint. We refer to [START_REF] Bubeck | Finite-time analysis of projected Langevin Monte Carlo[END_REF][START_REF] Lamperski | Projected Stochastic Gradient Langevin Algorithms for Constrained Sampling and Non-Convex Learning[END_REF] where W 1 -contraction properties from [START_REF] Eberle | Reflection couplings and contraction rates for diffusions[END_REF][START_REF] Wang | Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature[END_REF] are also used in an additive noise setting, but the multiplicative case is yet to be addressed by the literature.

Appendix

Lemma 2.11.1. Let Z and Z be two continuous diffusion processes. Then for all t ≥ 0 and for all p ≥ 2:

t 0 (σ(Z s ) -σ( Z s ))dW s p ≤ C BDG p [σ] Lip t 0 ∥Z s -Z s ∥ 2 p ds 1/2
, where C BDG p is a constant which only depends on p.

Proof. It follows from the generalized Minkowski and the Burkholder-Davis-Gundy inequalities that

t 0 (σ(Z s ) -σ( Z s ))dW s p ≤ C BDG p [σ] Lip t 0 |Z s -Z s | 2 ds 1/2 p/2 ≤ C BDG p [σ] Lip t 0 ∥Z s -Z s ∥ 2 p ds 1/2
We now give some results on the step sequence (γ n ) associated to the Euler-Maruyama scheme. Let us recall that the sequence (T n ) is defined in (2.4.7).

Lemma 2.11.2. Let (u n ) be a positive and non-increasing sequence such that n u n < ∞.

Then u n = o(n -1 ). Proof. We have N u 2N ≤ 2N n=N u n → 0 as N → ∞.
Lemma 2.11.3. We have

γ N (Tn) = o n -(1+β) .
(2.11.1)

Proof. Using the previous lemma,

γ n = o(n -1/2 ) so that Γ n = o(n 1/2 ) and then x 2 = o(N (x)) as x → ∞ and then γ N (Tn) = o N (T n ) -1/2 = o n -(1+β) . Lemma 2.11.4. The sequence (γ N (T n+1 -T ) /γ N (T n+1 ) ) is bounded.
Proof. Using (2.2.13, H γ2 ), we have for ϖ ′ > ϖ and for large enough k,

(γ k -γ k+1 )/γ 2 k+1 ≤ ϖ ′ so that γ k /γ k+1 ≤ 1 + ϖ ′ γ k+1 and then log γ N (T n+1 -T )) γ N (T n+1 ) = N (T n+1 )-1 k=N (T n+1 -T ) log γ k γ k+1 ≤ C N (T n+1 )-1 k=N (Tn) γ k = C Γ N (T n+1 ) -Γ N (T n+1 -T ) ≤ C(T n+1 -(T n+1 -T )).

Supplementary Material

Proof of Proposition 2.4.4

Proof. We have

ν a n+1 (x) ν an (x) = Z a n+1 Z an e -2(V (x)-V ⋆ )(a -2 n+1 -a -2 n ) ≤ Z a n+1 Z an =: M n .
We now consider

(P i ) 1≤i≤m ⋆ a partition of R d such that for all i, x ⋆ i ∈ Pi . Let us prove that for all 1 ≤ i ≤ m ⋆ , Z -1 a,i := R d e -2(V (x)-V ⋆ )/a 2 1 x∈P i dx ∼ a→0 a d R d e -x ⊤ ∇ 2 V (x ⋆ i )x dx.
(2.12.1) Let r > 0 ; let us consider V i defined as

V i (x) = V (x) if x ∈ B(x ⋆ i , r) |x -x ⋆ i | 2 + V ⋆ otherwise.
We also define

Z -1 a,i := R d e -2( V i (x)-V ⋆ )/a 2 1 x∈P i dx. Then, owing to V ⋆ > 0 and (2.2.8, H V 2 ), ∀x ∈ R d , C|x -x ⋆ i | 2 ≤ V i (x) -V ⋆ ≤ C ′ |x -x ⋆ i | 2
(2.12.2) and then

Z -1 a,i = a d R d e -2( V i (ax+x ⋆ i )-V ⋆ )/a 2 1 x∈a -1 (P i -x ⋆ i ) dx ∼ a→0 a d R d e -x ⊤ ∇ 2 V (x ⋆ i )x dx,
where we get the equivalence by dominated convergence ; the domination comes from (2.12.2). Then

Z -1 a,i -Z -1 a,i = B(x ⋆ i ,r) c e -2(V (x)-V ⋆ )/a 2 1 x∈P i dx - B(x ⋆ i ,r) c e -2( V i (x)-V ⋆ )/a 2 1 x∈P i dx =: I 1 -I 2 , I 2 = a d B(0,r/a) c e -2|x| 2 1 x∈a -1 (P i -x ⋆ i ) dx ≤ a d B(0,r/a) c e -2|x| 2 dx = o(a d ) = o Z -1 a,i .
Moreover using Proposition 4.3.1 we have Z a,i I 1 → 0 as a → 0, so that

Z -1 a,i = Z -1 a,i + o Z -1 a,i + o Z -1 a,i ∼ Z -1 a,i ,
which proves (2.12.1) and then

Z -1 a ∼ a→0 a d m ⋆ i=1 R d e -x ⊤ ∇ 2 V (x ⋆ i )x dx.
(2.12.3)

We now prove that

Z -1 an -Z -1 a n+1 ≤ Ca d-1 n+1 (a n -a n+1 ).
(2.12.4) Indeed, by convexity we have for all

z ∈ R e -2z/a 2 n -e -2z/a 2 n+1 ≤ 2e -2z/a 2 n z 1 a 2 n - 1 a 2 n+1 ≤ 4e -2z/a 2 n z a 2 n+1 (a n -a n+1 ) a n .
(2.12.5) and then

Z -1 an,i -Z -1 a n+1 ,i = a d n+1 R d e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n 1 x∈a -1 n+1 (P i -x ⋆ i ) -e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) dx ≤ 4a d-1 n+1 (a n -a n+1 ) R d e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n V (a n+1 x + x ⋆ i ) -V ⋆ a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) dx :=I 3 .
Let us also define

I 3 := R d e -2( V i (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n V i (a n+1 x + x ⋆ i ) -V ⋆ a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) dx.
Then I 3 converges by dominated convergence and |I 3 -

I 3 | is bounded by R d e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n V (a n+1 x + x ⋆ i ) -V ⋆ a 2 n+1 -e -2( V i (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n V i (a n+1 x + x ⋆ i ) -V ⋆ a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) dx ≤ a -d-2 n+1 B(x ⋆ i ,r) c e -2(V (x)-V ⋆ )/a 2 n (V (x) -V ⋆ )1 x∈P i dx + B(0,r/a n+1 ) c e -2( V i (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n V i (a n+1 x + x ⋆ i ) -V ⋆ a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) dx.
The second integral converges to 0 by dominated convergence by similar arguments as for I 2 . Moreover we have for every x ∈ B(x ⋆ i , r) c ∩ P i , V (x) -V ⋆ ≥ ε for some ε > 0 and then for n such that a n ≤ A/ √ 2:

a -d-2 n+1 B(x ⋆ i ,r) c e -2(V (x)-V ⋆ )/a 2 n (V (x) -V ⋆ )1 x∈P i dx ≤ Ca -d-2 n+1 B(x ⋆ i ,r) c e -2(V (x)-V ⋆ )/a 2 n |x -x ⋆ i | 2 1 x∈P i dx ≤ Ca -d-2 n+1 e -ε/a 2 n B(x ⋆ i ,r) c e -(V (x)-V ⋆ )/a 2 n |x -x ⋆ i | 2 1 x∈P i dx ≤ Ca -d-2 n+1 e -ε/a 2 n R d e -2(V (x)-V ⋆ )/A 2 |x -x ⋆ i | 2 dx -→ n→∞ 0,
where we used that

(x → |x| 2 e -2(V (x)-V ⋆ )/A 2 ) ∈ L 1 (R d ).
Then we obtain that I 3 converges to I 3 , which proves (2.12.4). Then we have

1 -M -1 n = Z -1 an -Z -1 a n+1 Z -1 an ≤ C a n -a n+1 a n ≤ C n log(n) .
On the other hand, if X ∼ ν a n+1 , X ∼ ν a n+1 , Y ∼ ν an and X, X and Y are mutually independent then

E|X -Y | -E|X -X| = a d n+1 Z an a d n+1 Z a n+1 m ⋆ i,j=1 a n+1 |x -y|e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 • e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n • 1 x∈a -1 n+1 (P i -x ⋆ i ) 1 y∈a -1 n+1 (P j -x ⋆ i ) dxdy -(a d n+1 Z a n+1 ) 2 m ⋆ i,j=1 a n+1 |x -y|e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 • e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) 1 y∈a -1 n+1 (P j -x ⋆ i ) dxdy = a 2d+1 n+1 Z a n+1 m ⋆ i,j=1 |x -y|e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 • Z an e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n -Z a n+1 e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n+1 • 1 x∈a -1 n+1 (P i -x ⋆ i ) 1 y∈a -1 n+1 (P j -x ⋆ i ) dxdy ≤ a n+1 a 2d n+1 Z 2 a n+1 m ⋆ i,j=1 |x -y|e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 • e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n -e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) 1 y∈a -1 n+1 (P j -x ⋆ i ) dxdy + a n+1 a 2d n+1 Z 2 a n+1 m ⋆ i,j=1 |x -y|e -2(V (a n+1 x+x ⋆ i )-V ⋆ )/a 2 n+1 e -2(V (a n+1 y+x ⋆ i )-V ⋆ )/a 2 n • 1 - Z an Z a n+1 1 x∈a -1 n+1 (P i -x ⋆ i ) 1 y∈a -1 n+1 (P j -x ⋆ i ) dxdy.
So using (2.12.5), dominated convergence as for the proof of (2.12.3), (2.12.3) itself with (2.4.9) and the bound for 1

-Z an /Z a n+1 = 1 -M -1 n we have lim sup n→∞ n log 3/2 (n) E|X -Y | -E|X -X| ≤ C m ⋆ i=1 |x -y|e -x ⊤ ∇ 2 V (x ⋆ i )x e -y ⊤ ∇ 2 V (x ⋆ i )y 1 + y ⊤ ∇ 2 V (x ⋆ i )y dxdy.
So that using Lemma 2.4.5 and the fact that E|X -X| is of order a n we have

W 1 (ν an , ν a n+1 ) ≤ E|X -Y | - 1 M n E|X -X| ≤ E|X -Y | -E|X -X| + C n log(n) E|X -X| ≤ C n log 3/2 (n) .
The proof for the second claim is similar.

Proof of Proposition 2.7.3

Proof. As in the proof of [PP23, Proposition 3.5], we split

|E[g( Ȳ x γ,u )] -E g(X x,n γ ) | into four terms A 1 , A 2 , A 3 and A 4 , that is, by the Taylor formula, for every y, z ∈ R d , g(z) -g(y) = ⟨∇g(y)|z -y⟩ + 1 0 (1 -u)∇ 2 g (uz + (1 -u)y) du(z -y) ⊗2 . For a given x ∈ R d , it follows that g(z) -g(y) = ⟨∇g(x)|z -y⟩ + ⟨∇g(y) -∇g(x)|z -y⟩ + 1 0 (1 -u)∇ 2 g (uz + (1 -u)y) (z -y) ⊗2 du = ⟨∇g(x)|z -y⟩ + ⟨∇ 2 g(x)(y -x)|z -y⟩ + 1 0 (1 -u)∇ 3 g(uy + (1 -u)x)(y -x) ⊗2 (z -y)du + 1 0 (1 -u)∇ 2 g (uz + (1 -u)y) du(z -y) ⊗2 .
Applying this expansion with y = X x,n γ and z = Ȳ x γ,u , this yields:

E[g( Ȳ x γ,u ) -g(X x,n γ )] = ⟨∇g(x)|E[ Ȳ x γ,u -X x,n γ ]⟩ =:A 1 + E ⟨∇ 2 g(x)(X x,n γ -x)| Ȳ x γ,u -X x,n γ ⟩ =:A 2 + E 1 0 (1 -u)∇ 3 g(uX x,n γ + (1 -u)x)(X x,n γ -x) ⊗2 ( Ȳ x γ,u -X x,n γ )du =:A 3 + 1 0 (1 -u)E ∇ 2 g u Ȳ x γ,u + (1 -u)X x,n γ ( Ȳ x γ,u -X x,n γ ) ⊗2 du =:A 4 . • Term A 1 : The term A 1 is bounded by |∇g(x)| • |E[ Ȳ x γ,u -X x,n γ ]|, with E[ Ȳ x γ,u -X x,n γ ] = E γ 0 b a(u) (x) -b a(u) (X x,n s ))ds + E γ 0 (b a(u) (X x,n s ) -b a n+1 (X x,n s ))ds =: A 11 + A 12 .
We have

|A 12 | ≤ γ||Υ|| ∞ (a 2 n -a 2 n+1 )
and

|A 11 | = γ 0 s 0 E ∇b a(u) (X x,n v )b a(u) (X x,n v ) + 1 2 ∇ 2 b a(u) (X x,n v )a 2 n+1 σσ ⊤ (X x,n v ) dv ≤ Cγ 2 sup v∈[0,γ] E[V 1/2 (X x,n v )] ≤ Cγ 2 V 1/2 (x),
where we used that

|∇b a | ≤ C and ∥∇ 2 b a ∥ ≤ CV 1/2 because we assumed ∥∇ 3 V ∥ ≤ CV 1/2 and ∥∇ 3 (σσ ⊤ )∥ ≤ CV 1/2 .
• Term A 2 : We have:

|A 2 | ≤ 1≤i,j≤d |∂ ij g(x)||E[(X x,n γ -x) i (X x,n γ -Ȳ x γ,u ) j ]|
and we have

E[(X x,n γ -x) i (X x,n γ -Ȳ x γ,u ) j ] = E[(X x,n γ -Ȳ x γ,u ) i (X x,n γ -Ȳ x γ,u ) j ] + E[( Ȳ x γ,u -x) i (X x,n γ -Ȳ x γ,u ) j ].
Using Lemma 2.7.2, the first term of the right-hand side is bounded by

C(V 1/2 (x)γ + √ γ(a n - a n+1
)) 2 and in the second term we write ( Ȳ x γ,u -

x) i = γb a(u) (x) + γζ k+1 (x) + a(u)σ(x)W γ i and we have |E[(γb a(u) (x) + γζ k+1 (x)) i (X x,n γ -Ȳ x γ,u ) j ]| ≤ γV 1/2 (x)(V 1/2 (x)γ + √ γ(a n -a n+1 ))
and using that the increments of ζ and W are independent,

|E[(a(u)σ(x)W γ ) i (γζ k+1 (x)) j ]| = 0
and using the Itō isometry:

E (a(u)σ(x)W γ ) i γ 0 (b a n+1 (X x,n s ) -b a n+1 (x) + b a n+1 (x) -b a(u) (x)) j ds + γ 0 ((a n+1 σ(X x,n s ) -a n+1 σ(x) + a n+1 σ(x) -a(u)σ(x))dW s ) j ≤ C[b] Lip γ 0 ||W γ || 2 ||X x,n s -x|| 2 ds + C(a 2 n -a 2 n+1 )||W γ || 1 γ||Υ|| ∞ + C d k=1 γ 0 E[σ ik (x)(σ jk (X x,n s ) -σ jk (x)]ds + C(a n -a n+1 )E[W 2 γ ] ≤ CV 1/2 (x)γ 2 + C(a n -a n+1 )γ 3/2 + CV 1/2 (x)γ 2 + C(a n -a n+1 )γ,
where we used an argument similar to A 11 to bound the third term, using that ∇σ and ∇ 2 σ are bounded.

• Term A 3 : Using the three fold Cauchy-Schwarz inequality, Lemma 2.6.3 and Lemma 2.7.2, A 3 is bounded by

C sup ξ∈(x,X x,n γ ) ||∇ 3 g(ξ)|| 4 V (x)γ V 1/2 (x)γ + √ γ(a n -a n+1 ) . • Term A 4 : Using Lemma 2.7.2, A 4 is bounded by C V 1/2 (x)γ + √ γ(a n -a n+1 ) 2 sup ξ∈(X x,n γ , Ȳ x γ,u ) ||∇ 2 g(ξ)|| 2 .

Proof of Theorem 2.2.4

Proof. We remark that according to Proposition 4.3.1, for all κ > 0 we have e -κg ∈ L 1 (R d ). We first prove that

W 1 (ν an , ν a n+1 ) ≤ C n log 1+α min (n) ,
(2.12.6) so that (W 1 (ν an , ν a n+1 )) is still a converging Bertrand series. To do so, we directly adapt the proof of Proposition 2.4.4, replacing the change of variables in the integrals in ax by the change of variables in B • (a 2α 1 x 1 , . . . , a 2α d x d ). Still using (2.12.5), we successively obtain

Z -1 a ∼ a→0 a 2α 1 +•••+2α d R d e -2g(x) dx Z -1 an -Z -1 a n+1 ≤ 4a 2α 1 +•••+2α d -1 n+1 (a n -a n+1 ) R d e -2g(x) g(x)dx 1 -M -1 n ≤ C n log(n) E|X -Y | -E|X -X| ≤ Ca 2α min n+1 n log(n) .
Then, using (2.12.6) we prove that W 1 (ν n , ν ⋆ ) ≤ Ca 2α min n the same way as in Lemma 2.4.6. The next parts of the proof are the same as for the definite positive case.

Proof of Theorem 2.5.1

To prove Theorem 2.8.1, we proceed as for the proof of Theorem 2.2.1.

In the following, for γ > 0 we denote by ( Xx,n,γ t

) t∈[0,γ] the Euler-Maruyama scheme over one step with coefficient a n+1 . We first recall [PP23, Lemma 3.4(b), Proposition 3.5(a)] giving bounds for the weak and strong errors for the one-step Euler-Maruyama scheme, which do not depend on the ellipticity parameter a n . Lemma 2.12.1. Let p ≥ 1 and let γ > 0. There exists C > 0 such that for every n ≥ 0, for every γ ∈ (0, γ] and every t ∈ [0, γ]:

||X x,n t -Xx,n,γ t || p ≤ CV 1/2 (x)t.
Proposition 2.12.2. Let γ > 0. Then for every g : R d → R being C 3 and for every 0

≤ γ ≤ γ ′ ≤ γ: E g( Xx,n,γ ′ γ ) -E g(X x,n γ ) ≤ CV 3/2 (x)γ 2 Φ g (x)
, where

Φ g (x) = max   |∇g(x)|, ||∇ 2 g(x)||, sup ξ∈(X x,n γ , Xx,n,γ ′ γ ) ||∇ 2 g(ξ)|| 2 , sup ξ∈(x,X x,n γ ) ||∇ 3 g(ξ)|| 4   .
Proposition 2.12.3. Let T , γ > 0. Then for every Lipschitz continuous function f : R d → R, for every n ≥ 0 and every t ∈ (0, T ] and every 0 ≤ γ ≤ γ ′ ≤ γ:

E P t f ( Xx,n,γ ′ γ ) -E P t f (X x,n γ ) ≤ Ca -3 n [f ] Lip γ 2 t -1 V 2 (x). Proof. The proof is the same as in [PP23, Proposition 3.6]. When applying [PP23, Proposition 3.2(b)],
we remark that the lowest exponent of σ0 is -3. Moreover, by the same proof as in Lemma 2.7.1 we get sup m≥k+1 EV p ( Xx,n

Γm-Γ k ) ≤ CV p (x). We now prove Theorem 2.8.1.

Proof. Let us write:

W 1 ([ Xx 0 Tn ], ν ⋆ ) ≤ W 1 ([ Xx 0 Tn ], [X x 0 Tn ]) + W 1 ([X x 0 Tn ], ν ⋆ ). Temporarily setting xn := Xx 0
Tn and x n := X x 0 Tn , we have

W 1 ([ Xx 0 T n+1 ], [X x 0 T n+1 ]) = W 1 ([ X xn,n T n+1 -Tn ], [X xn,n T n+1 -Tn ]) ≤ W 1 ([ X xn,n T n+1 -Tn ], [X xn,n T n+1 -Tn ]) + W 1 ([X xn,n T n+1 -Tn ], [X xn,n T n+1 -Tn ]
), and we find a bound on the first term using the same proof as in [PP23, Section 4.2]. For

x ∈ R d , we split |Ef ( Xx,n T n+1 -Tn ) -Ef (X x,n
T n+1 -Tn )| into three terms (a), (b) and (c). We however pay attention to the dependence in a n when applying Lemma 2.12.1, Proposition 2.12.3 and Theorem 2.4.2. We then have:

(c) ≤ C[f ] Lip γ N (Tn) V 1/2 (x), (b) ≤ Ca -3 n+1 γ N (Tn) log T + ||γ|| ∞ γ N (Tn) , (a) ≤ Ca -3 n+1 e C 1 a -2 n+1 V (x)γ N (Tn) ρ -1 n+1 .
Then we establish a recursive relation and prove the convergence as in the proof of Theorem 2.2.1.

Chapter 3

Convergence of Langevin-Simulated Annealing algorithms with multiplicative noise II: Total Variation

Introduction

Langevin-based algorithms are used to solve optimization problems in high dimension and have gained much interest in relation with Machine Learning. The Langevin equation is a Stochastic Differential Equation (SDE) which consists in a gradient descent with noise. More precisely, let V : R d → R + be a coercive potential function, then the associated Langevin equation reads

dX t = -∇V (X t )dt + σdW t , t ≥ 0,
where (W t ) is a d-dimensional Brownian motion and where σ > 0. Under standard assumptions, the invariant measure of this SDE is the Gibbs measure ν σ 2 of density proportional to e -2V (x)/σ 2 and for small enough σ, this measure concentrates around argmin(V ) see [START_REF] Arnak | Theoretical guarantees for approximate sampling from smooth and log-concave densities[END_REF] and Chapter 4. Adding a small noise to the gradient descent allows to explore the space and to escape from traps such as local minima or saddle points appearing in non-convex optimization problems [Laz92, DPG + 14]. Such methods have been recently brought up to light again with Stochastic Gradient Langevin Dynamics (SGLD) algorithms [START_REF] Welling | Bayesian Learning via Stochastic Gradient Langevin Dynamics[END_REF][START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF], especially for the deep learning and the calibration of large artificial neural networks.

The Langevin-simulated annealing SDE is the Langevin equation where the noise parameter is slowly decreasing to 0, namely

dX t = -∇V (X t )dt + a(t)σdW t , t ≥ 0, (3.1.1)
where a : R + → R + is non-increasing and converges to 0. The idea is that the "instantaneous" invariant measure ν a(t)σ which is the Gibbs measure of density ∝ exp(-2V (x)/(a(t) 2 σ 2 )) converges itself to argmin(V ). Although the additive case i.e. where σ is constant has been extensively studied (see for example [DM17, DM19]), little attention has been paid to the multiplicative case i.e. where σ :

R d → M d (R) depends on X t .
The objective of the present paper is to study the convergence in total variation of the Langevin-Simulated annealing SDE, i.e. (3.1.1) with non-constant σ. Following [PP23, Proposition 2.6], we need to add a correction term in the drift, giving

dY t = -(σσ ⊤ ∇V )(Y t )dt + a(t)σ(Y t )dW t +   a 2 (t)   d j=1 ∂ j (σσ ⊤ )(Y t ) ij   1≤i≤d   dt, (3.1.2) a(t) = A log(t + e) , (3.1.3)
so that ν a(t) is still the the "instantaneous" invariant measure. We also study the convergence of its Euler-Maruyama scheme Ȳt with decreasing steps and with noisy gradient estimates coming from stochastic gradient algorithms. We assume in particular the convex uniformity of the potential V outside a compact set (but we do not assume that the potential is convex) and the ellipticity and the boundedness of σ.

We studied this SDE and proved the convergence in L 1 -Wasserstein distance of Y and Ȳ to ν ⋆ which is the limit measure of ν a as a → 0, in a previous chapter, see Chapter 2. More precisely, we proved that

W 1 (Y t , ν ⋆ ) is of order a(t) as t → ∞ and that W 1 (Y t , ν (a(t))
) is of order t -α for every α ∈ (0, 1). For more details, we refer to the introduction of Chapter 2. In particular, for applications to optimization problems arising in Stochastic Optimization and in Machine Learning and for choices of σ : R d → M d (R) used by practitioners, we refer to Section 2.3.

The proof for the total variation distance case relies on the same strategy developed in Chapter 2. We first introduce the process X where the coefficient (a(t)) is "by plateaux" i.e. non-increasing and piecewise constant on time intervals [T n , T n+1 ]. Then we give bounds on d TV (X t , Y t ) using a domino strategy (2.1.4). However the main difference with the L 1 -Wasserstein distance concerns the total variation distance between X and Y in small time as in general, it is more difficult to give bounds in small time for the total variation distance between two processes with close coefficients. Indeed, considering the functional characterization and comparing it with the L 1 -Wasserstein distance, if x and y ∈ R d are close to each other and if

f : R d → R is Lipschitz-continuous, then we can bound |f (x) -f (y)| by [f ] Lip |x -y|; however if f is measurable bounded, then we cannot directly bound |f (x) -f (y)| in terms of |x -y|.
Instead, the common strategy of proof in the literature is to use Malliavin calculus in order to perform an integration by parts and to use bounds on the derivatives of the density. In this context, [START_REF] Pagès | Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds[END_REF] relies on a highly technical Malliavin approach inducing a "regularization from the past" (see [PP23, Theorem 3.7 and Appendix B]).

We give bounds in small time relying on Chapter 8 and we adapt some of the proofs to the non-homogeneous Markovian setting. These bounds rely on estimates of the density of the solutions to SDE's and their derivatives [START_REF] Friedman | Partial differential equations of parabolic type[END_REF]. The strategy of proof is the following: we first reduce to the null drift case using a Girsanov change of measure. Then we introduce an artificial regularization in order to perform a Malliavin-type integration by parts and we use Aronson's bounds on the density and its derivatives; we need to pay attention to the dependency in the parameter a, controlling the ellipticity of the SDE and which converges to 0, of the constants that appear in the Aronson bounds. Moreover, we rely on [START_REF] Devroye | The total variation distance between high-dimensional Gaussians[END_REF] to give bounds on the total variation between two Gaussian laws.

Contrary to the L 1 -Wasserstein distance, we do not prove the convergence as t → ∞ of Y t and Ȳt to ν ⋆ since in most of the cases, ν ⋆ is supported by a finite number of points and then if Y t has a density then d TV (Y t , ν ⋆ ) = 2. Instead, we prove the convergence in total variation of Y t and Ȳt to their "instantaneous invariant measure" ν a(t) which itself converges to ν ⋆ (in law, for the L 1 -Wasserstein distance etc, see for example [Hwa80, Theorem 2.1] and Lemma 2.4.6 and we give bounds on d TV (Y t , ν a(t) ) and on d TV ( Ȳt , ν a(t) ) as t → ∞.

The paper is organized as follows. In Section 3.2 we give the setting and assumptions of the problem we consider and state our main results of convergence with convergence rates. This setting is the same as in Chapter 2. In Section 3.3 we establish bounds in small time for d TV (X t , Y t ) and for d TV (X t , Ȳt ), inspired from Chapter 8. In Section 3.4, we prove the convergence of the plateaux SDE X using exponential contraction properties. Using this convergence, the convergences of d TV (Y t , ν a(t) ) and d TV ( Ȳt , ν a(t) ) are proved in Section 3.5 and 3.6 respectively. In section 3.7 we compare additive and multiplicative Langevin algorithms on a numerical optimization problem and we give numerical evidence that multiplicative Langevin algorithms improve the optimization procedure.

We use the notations defined from page 1.

Assumptions and main results

Assumptions

Let us briefly recall the setting adopted in Chapter 2. Let V : R d → (0, +∞) be a C 2 potential function such that V is coercive and

(x → |x| 2 e -2V (x)/A 2 ) ∈ L 1 (R d ) for some A > 0. (3.2.1)
Then V admits a minimum on R d . Moreover, let us assume that

V ⋆ := min R d V > 0, argmin(V ) = {x ⋆ 1 , . . . , x ⋆ m ⋆ }, ∀ i = 1, . . . , m ⋆ , ∇ 2 V (x ⋆ i ) > 0, (3.2.2, H V 1 )
i.e. min R d V is attained at a finite number m ⋆ of points and at each point the Hessian matrix is positive definite. We then define for a ∈ (0, A] the Gibbs measure ν a of density :

ν a (dx) = Z a e -2(V (x)-V ⋆ )/a 2 dx, Z a = R d e -2(V (x)-V ⋆ )/a 2 dx -1 (3.2.3)
Following [Hwa80, Theorem 2.1], the measure ν a converges weakly to ν ⋆ as a → 0, where ν ⋆ is the weighted sum of Dirac measures:

ν ⋆ =   m ⋆ j=1 det ∇ 2 V (x ⋆ j ) -1/2   -1 m ⋆ i=1 det ∇ 2 V (x ⋆ i ) -1/2 δ x ⋆ i . (3.2.4)
Following Lemma 2.4.6, ν a also converges to ν ⋆ as a → 0 for the L 1 -Wasserstein distance.

We consider the following Langevin SDE in R d :

Y x 0 0 = x 0 ∈ R d , dY x 0 t = b a(t) (Y x 0 t )dt + a(t)σ(Y x 0 t )dW t , (3.2.5)
where, for a ≥ 0, the drift b a is given by

b a (x) = -(σσ ⊤ ∇V )(x) + a 2   d j=1 ∂ j (σσ ⊤ ) ij (x)   1≤i≤d =: -(σσ ⊤ ∇V )(x) + a 2 Υ(x), (3.2.6)
where W is a standard R d -valued Brownian motion defined on a probability space (Ω, A, P), where σ :

R d → M d (R) is C 2 and a(t) = A log(t + e) (3.2.7)
where A is defined in (3.2.1) and with log(e) = 1. This equation corresponds to a gradient descent on the potential V with preconditioning σ and multiplicative noise ; the second term in the drift (3.2.6) is a correction term (see [PP23, Proposition 2.6]) which is zero for constant σ.

We make the following assumptions on the potential V :

|∇V | 2 ≤ CV and sup x∈R d ||∇ 2 V (x)|| < +∞, (3.2.8, H V 2 )
which implies in particular that V has at most a quadratic growth. Let us also assume that

σ is bounded and Lipschitz-continuous, ∇ 2 σ is bounded, ∇(σσ ⊤ )∇V is bounded, (3.2.9, H σ )
and that σ is uniformly elliptic, i.e.

∃ σ0 > 0, ∀x ∈ R d , (σσ ⊤ )(x) ≥ σ2 0 I d . (3.2.10) Assumptions (3.2.8, H V 2
) and (3.2.9, H σ ) imply that Υ is also bounded and Lipschitz-continuous and that b a is Lipschitz-continuous uniformly in a

∈ [0, A]. Let the minimal constant [b] Lip be such that: ∀a ∈ [0, A], b a is [b] Lip -Lipschitz continuous. (3.2.11)
We make the non-uniform dissipative (or convexity) assumption outside of a compact set: there exists α 0 > 0 and R 0 > 0 such that

∀x, y ∈ B(0, R 0 ) c , σσ ⊤ ∇V (x) -σσ ⊤ ∇V (y), x -y ≥ α 0 |x -y| 2 . (3.2.12, H cf )
Taking y ∈ B(0, R 0 ) c fixed, letting |x| → ∞ and using the boundedness of σ, (3.2.12, H cf ) implies that |∇V | is coercive. Using (3.2.8, H V 2 ) and the boundedness of σ, there exists C > 0 (depending on A) such that:

∀a ∈ [0, A], 1 + |b a (x)| ≤ CV 1/2 (x).
Let (γ n ) n≥1 be a non-increasing sequence of varying positive steps. We define Γ n := γ 1 + • • • + γ n and for t ≥ 0:

N (t) := min{k ≥ 0 : Γ k+1 > t} = max{k ≥ 0 : Γ k ≤ t}.
(3.2.13)

We make the classical assumptions on the step sequence, namely

γ n ↓ 0, n≥1 γ n = +∞ and n≥1 γ 2 n < +∞ (3.2.14, H γ1 )
and we also assume that

ϖ := lim sup n→∞ γ n -γ n+1 γ 2 n+1 < ∞. (3.2.15, H γ2 ) For example, if γ n = γ 1 /n η with η ∈ (1/2, 1) then ϖ = 0; if γ n = γ 1 /n then ϖ = γ 1 .
In stochastic gradient algorithms, the true gradient is measured with a zero-mean noise ζ, which law only depends on the current position. That is, let us consider a family of random fields (

ζ n (x)) x∈R d ,n∈N such that for every n ∈ N, (ω, x) ∈ Ω × R d → ζ n (x, ω) is measurable
and for all x ∈ R d , the law of ζ n (x) only depends on x and (ζ n (x)) n∈N is an i.i.d. sequence independent of W . We make the following assumptions:

∀x ∈ R d , ∀p ≥ 1, E[ζ 1 (x)] = 0 and E[|ζ 1 (x)| p ] ≤ C p V p/2 (x).
(3.2.16)

We then consider the Euler-Maruyama scheme with decreasing steps associated to (Y t ):

Ȳ x 0 0 = x 0 , Ȳ x 0 Γ n+1 = ȲΓn + γ n+1 b a(Γn) ( Ȳ x 0 Γn ) + ζ n+1 ( Ȳ x 0 Γn ) + a(Γ n )σ( Ȳ x 0 Γn )(W Γ n+1 -W Γn ), (3.2.17)
We extend Ȳ x 0

•

on R + by considering its genuine continuous interpolation:

∀t ∈ [Γ n , Γ n+1 ), Ȳ x 0 t = Ȳ x 0 Γn + (t -Γ n ) b a(Γn) ( Ȳ x 0 Γn ) + ζ n+1 ( Ȳ x 0 Γn ) + a(Γ n )σ( Ȳ x 0 Γn )(W t -W Γn ). (3.2.18)

Main results

Theorem

3.2.1. (a) Let Y be defined in (3.2.5). Assume (3.2.2, H V 1 ), (3.2.8, H V 2 ), (3.2.9, H σ ), ( 3
.2.10) and (3.2.12, H cf ). Then, for every α ∈ (0, 1), if A is large enough, then for every x 0 ∈ R d and for every t > 0:

d TV Y x 0 t , ν a(t) ≤ Ce C √ log(t)(1+|x 0 | 2 ) t -α . (3.2.19) (b) Let Ȳ be defined in (3.2.17). Assume (3.2.2, H V 1 ), (3.2.8, H V 2 ), (3.2.9, H σ ), (3.2.10) and (3.2.12, H cf ). Assume furthermore that σ ∈ C 2r b . Assume furthermore (3.2.14, H γ1 ) and (3.2.15, H γ2 ), that V is C 3 with ∥∇ 3 V ∥ ≤ CV 1/2 and that σ is C 3 with ∥∇ 3 (σσ ⊤ )∥ ≤ CV 1/2
. Then, for every α ∈ (0, 1), if A is large enough, then for every x 0 ∈ R d and for every t > 0:

d TV Ȳ x 0 t , ν a(t) ≤ C log 1/2 (t) max V 2 (x 0 ), 1 + |x 0 | t -α + e C √ log(t)(1+|x 0 | 2 ) t C/A 2 γ r/(2r+1) N (Ct)
.

(3.2.20)

Remark 3.2.2. Depending on the step sequence (γ n ), we can compare the two terms arising in the right-hand side of (3.2.20). For example, if

γ n = γ 1 n -η for some η ∈ (1/2, 1], then • If η = 1, then γ N (Ct) ≍ e -Ct
and the first term is the dominating term.

• If η ∈ (1/2, 1) then γ N (Ct) ≍ (Ct) -η/(1-η) .

Extensions and interpolations of the processes

Let us define the following processes that will be used as auxiliary tools in the proofs.

• We define (X t ) as the solution the following SDE where the coefficients piecewisely depend on the time; X is then said to be "by plateaux":

X x 0 0 = x 0 , dX x 0 t = b a k+1 (X x 0 t )dt + a k+1 σ(X x 0 t )dW t , t ∈ [T k , T k+1 ], (3.2.21)
where b a is defined in (3.2.6) and the time schedule (T n ) is defined by

T n := C (T ) n 1+β , (3.2.22)
where C (T ) > 0, β > 0 and a n := a(T n ). More generally, we define (X x,n t ) as the solution of

X x,n 0 = x, dX x,n t = b a k+1 (X x,n t )dt + a k+1 σ(X x,n t )dW t , t ∈ [T k -T n , T k+1 -T n ], k ≥ n, (3.2.23) i.e. (X x,n t ) has the conditional law of (X Tn+t ) t≥0 given X Tn = x. We have X x t = X x,0 t . The Markov transition kernel associated to X •,n denoted P X,n t reads on Borel functions f : R d → R + , P X,n t f (x) = E[f (X x,n t )].
• Considering now the original SDE (3.2.5), we also define for every x ∈ R d and every fixed u ≥ 0:

Y x 0,u = x, dY x t,u = b a(t+u) (Y x t,u )dt + a(t + u)σ(Y x t,u )dW t , (3.2.24) so that Y x = Y x •,0 .
We define the Markov transition kernel associated to Y between the times t and t + u by P Y t,u such that for all Borel functions f :

R d → R + , P Y t,u f (x) = E[f (Y x t,u )].
• Considering finally (3.2.17) and (3.2.18), we define for every n ≥ 0, ( Ȳ x t,Γn ) t≥0 , first at times

Γ k -Γ n , k ≥ n, by Ȳ x 0,Γn = x, Ȳ x Γ k+1 -Γn,Γn = Ȳ x Γ k -Γn,Γn + γ k+1 b a(Γ k ) ( Ȳ x Γ k -Γn,Γn ) + ζ k+1 ( Ȳ x Γ k -Γn,Γn ) + a(Γ k )σ( Ȳ x Γ k -Γn,Γn )(W Γ k+1 -W Γ k ), (3.2.25)
then at every time t by the genuine interpolation on the intervals

([Γ k -Γ n , Γ k+1 -Γ n )) k≥n as before. In particular Ȳ x = Ȳ x •,0 . Still more generally, we define Ȳ x t,u where u ∈ (Γ n , Γ n+1 ) as Ȳ x 0,u = x, Ȳ x t,u =    x + t(b a (x) + ζ n+1 (x)) + a 2 (u)σ(x)(W t -W Γu ) if t ∈ [u, Γ n+1 ] = Ȳ Ȳ x Γ n+1 -u,u t-(Γ n+1 -u),Γ n+1 if t > Γ n+1 .
For n, k ≥ 0, for u ∈ [Γ k , Γ k+1 ) and γ ∈ [0, Γ k+1 -u], let P Ȳ γ,u be the Markov transition kernel associated to Ȳ•,u between the times 0 and γ i.e. for all Borel functions f :

R d → R + , P Ȳ γ,u f (x) = E[f ( Ȳ x γ,u )].

Bounds in total variation for small t

In this section we give bounds for the total variation distance between the processes X, Y and Ȳ . Although such bounds are straightforward for L p -distances, they are more difficult to establish for d TV . To this end we adopt a strategy similar to Chapter 8. For x ∈ R d and for a ∈ R + we define the "cut" drift bx a : R d → R d which is the drift b a which is null outside a compact set centred on x. More precisely, we choose R > 0 and we consider a C ∞ decreasing function ψ : R + → R + such that ψ = 1 on [0, R 2 ] and ψ = 0 on [(R + 1) 2 , ∞) and we define bx

a (y) := b a (y)ψ(|y -x| 2 ), so that | bx a | is bounded by C(1 + |x|) since b a is Lipschitz-continuous.
For σ : R d → M d (R), we denote the martingale:

M (σ) x 0 = x, dM (σ) x t = σ(M (σ) x t )dW t (3.3.1)
with its associated one-step Euler-Maruyama scheme:

M (σ) x t = x + σ(x)W t . (3.3.2) Lemma 3.3.1.
Let Z be solution of the following SDE:

dZ x t = u(t)σ Z (Z x t )dW t ,
where u :

R + → (0, ∞) is C 1 and bounded. Then (Z t ) ∼ (M (σ) F (-1) (t) )
, where F : R + → R + is solution of the differential equation

F (0) = 0, F ′ (t) = 1 u 2 (F (t))
and where F (-1) denotes the (continuous) inverse function of F . Proof. First, F is well defined and is strictly increasing with F (t) → ∞ as t → ∞ since u is bounded, so that F (-1) : R + → R + is well defined as well. We have

d Z x F (t) = u(F (t))σ Z (Z x F (t) )d W F (t) = F ′ (t) 1/2 u(F (t))σ Z (Z x F (t) )d W t = σ Z (Z x F (t) )d W t .
where W is the Brownian motion defined by W t = t 0 (F ′ (s)) -1/2 dW F (s) .

Total variation bound in small time for the Euler-Maruyama scheme

Proposition 3.3.2. Assume that σ ∈ C 2r b . There exists C > 0 such that for every n, k ≥ 0, for every u ∈ [Γ k , Γ k+1 ) and every t > 0 such that u ∈ [T n , T n+1 ], t ≤ Γ k+1 -u and u+t ∈ [T n , T n+1 ], d TV (X x,n t , Ȳ x t,u ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t r/(2r+1) + Ca -2 n (a n -a n+1 ). (3.3.3)
Proof. We apply a strategy of proof similar to Theorem 8.2.2. However we need to pay attention to the dependency of the constants in the bounds in (a n ). Let us write

d TV (X x,n t , Ȳ x t,u ) ≤ d TV (X x,n t , X x,n t ) + d TV ( X x,n t , Z x,n t ) + d TV (Z x,n t , Zx,n t ) + d TV ( Zx,n t , Xx,n t ) + d TV ( Xx,n t , Ȳ x t,u ), (3.3.4)
where

X x,n 0 = x, d X x,n t = bx a n+1 ( X x,n t )dt + a n+1 σ( X x,n t )dW t , Z x,n 0 = x, dZ x,n t = a n+1 σ(Z x,n t )dW t , Zx,n 0 = x, Zx,n t = x + a n+1 σ(x)W t . • Using Lemma 8.3.3, we have d TV (X x,n t , X x,n t ) ≤ C(1 + |x| 2 )t,
where the constant C does not depend on n.

• We use [QZ04, Theorem 2.4] and we rework the bound from Lemma 8.3.6 to make explicit the dependency in a n . Reworking Lemma 8.3.5, we have for q ≥ 1:

E sup s∈[0,t] |U x,n s | 2q ≤ Ce Cqa -1 n+1 (1+|x| 2 ) , (3.3.5) U x,n 0 = 1, dU x,n s = a -1 n+1 U x,n s σ -1 (Z x,n s ) bx a n+1 (Z x,n s ), dW s .
Moreover, following Lemma 3.3.1 we have (Z x,n t ) ∼ (M (σ) x F (-1) (t) ) where the process (M (σ) t ) does not depend on n and where F (-1) (t) = a 2 n+1 t. Thus following [Fri64, Chapter 9, Theorem 7] (also see Theorem 8.3.1 for the application to SDE's) and since σ ∈ C 2 b we have

|∇ x p M (σ) (t, x, y)| ≤ C t (d+1)/2 e -c|y-x| 2 /t (3.3.6)
and then

|∇ x p Z (t, x, y)| = |∇ x p M (σ) (a 2 n+1 t, x, y)| ≤ Ca -(d+1) n+1 t (d+1)/2 e -ca -2 n+1 |y-x| 2 /t ≤ Ca -(d+1) n+1 t (d+1)/2 e -c|y-x| 2 /t .
Then using Lemma 8.3.6 with the adapted bound on U x,n s (3.3.5) along with [QZ04, Theorem 2.4], we obtain d

TV ( X x,n t , Z x,n t ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 . (3.3.7)
The same way, we obtain

d TV ( Zx,n t , Xx,n t ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 . • Following Lemma 3.3.1, we have (Z x,n t ) ∼ (M (σ) x a 2 n+1 t ) and ( Zx,n t ) ∼ ( M (σ) x a 2 n+1 t )
, where both processes M (σ) and M (σ) do not depend on n. We then use Theorem 8.2.2 to get

d TV (M (σ) x t , M (σ) x t ) ≤ Ce C|x| 2 t r/(2r+1) which implies d TV (Z x,n t , Zx,n t ) ≤ Ce C|x| 2 a 2r/(2r+1) n+1 t r/(2r+1) ≤ Ce C|x| 2 t r/(2r+1
∼ N x + tb a n+1 (x), a 2 n+1 tσσ ⊤ (x) , Ȳ x t,u ∼ N x + tb a(u) (x) + tζ(x), a 2 (u)tσσ ⊤ (x) .
Then, conditionally to ζ(x) we have

d TV ( Xx,n t , Ȳ x t,u ) ≤ d TV N x + tb a n+1 (x), a 2 n+1 tσσ ⊤ (x) , N x + tb a(u) (x) + tζ(x), a 2 n+1 tσσ ⊤ (x) + d TV N x + tb a(u) (x) + tζ(x), a 2 n+1 tσσ ⊤ (x) , N x + tb a(u) (x) + tζ(x), a 2 (u)tσσ ⊤ (x) =: D 1 + D 2 .
We then refer to [START_REF] Devroye | The total variation distance between high-dimensional Gaussians[END_REF] which gives bounds on the total variation between two Gaussian laws, first in the case

d > 1. Using [DMR18, Theorem 1.1] with λ 1 = • • • = λ d = (a(u) 2 -a 2 n+1 )/a 2 n+1 , we have D 2 ≤ C a 2 (u) -a 2 n+1 a 2 n+1 ≤ Ca -1 n (a n -a n+1 ).
Using [DMR18, Theorem 1.2], since the ρ i 's are bounded independently of n and since for every

y ∈ R d , y ⊤ σσ ⊤ (x)y ≥ σ2 0 |y| 2 , we have D 1 ≤ C √ ta -1 n+1 (1 + |ζ(x)| 1/2 ).
Now, integrating over the law of ζ(x) and using that E|ζ(x)| ≤ CV (x), we obtain

d TV ( Xx,n t , Ȳ x t,u ) ≤ Ca -1 n (a n -a n+1 ) + C √ t(1 + V 1/2 (x)).
In the case d = 1, we use [DMR18, Theorem 1.3] and obtain the same bounds.

• Conclusion:

Considering (3.3.4), we get d TV (X x,n t , Ȳ x t,u ) ≤ C(1 + |x| 2 )t + Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 + Ce C|x| 2 t r/(2r+1) + Ca -1 n (a n -a n+1 ) + C √ t(1 + V 1/2 (x)) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t r/(2r+1) + Ca -1 n (a n -a n+1 ).

Total variation bound in small time for the continuous SDE

Proposition 3.3.3. Assume that σ ∈ C 2r b and let γ > 0. There exists C > 0 such that for all

ε > 0, n ≥ 0, u, t ≥ 0 such that u ∈ [T n , T n+1 ], u + t ∈ [T n , T n+1 ] and t ≤ γ, d TV (X x,n t , Y x t,u ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 + Ca -(d+r) n+1 (a(u) -a n+1 ) 2r/(2r+1) . (3.3.8) Proof. We have d TV (X x,n t , Y x t,u ) ≤ d TV (X x,n t , X x,n t ) + d TV ( X x,n t , Z x,n t ) + d TV (Z x,n t , Z x t,u ) + d TV ( Z x t,u , Y x t,u ) + d TV ( Y x t,u , Y x t,u ) (3.3.9) where d X x,n t = bx a n+1 ( X x,n t )dt + a n+1 σ( X x,n t )dW t , dZ x,n t = a n+1 σ(Z x,n t )dW t , d Z x t,u = a(u + t)σ( Z x t,u )dW t , d Y x t,u = bx a(u+t) ( Y x t,u )dt + a(u + t)σ( Y x t,u )dW t .
Using Lemma 8.3.3, we have

d TV (X x,n t , X x,n t ) + d TV ( Y x t,u , Y x t,u ) ≤ C(1 + |x| 2 )t.
Using (3.3.7) again, we have

d TV ( X x,n t , Z x,n t ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 .
Moreover, using [QZ04, Theorem 2.4] (with an immediate adaptation to the non-homogeneous case) and establishing the same bounds as in Lemma 8.3.6, we also have

d TV ( Z x t,u , Y x t,u ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 . (3.3.10)
We now turn to d TV (Z x,n t , Z x t,u ). Using Lemma 3.3.1 as in (3.3.6) we have

|∇ 2r y p Z (t, x, y)| = |∇ 2r y p M (σ) (a 2 n+1 t, x, y)| ≤ C a -(d+r) n+1 t (d+r)/2 e -c|x-y| 2 /t .
To bound p Z we use the change of time

F satisfying F ′ (t) = a -2 (u + F (t)) so that a -2 n t ≤ F (t) ≤ a -2 n+1 t and a 2 n+1 t ≤ F (-1) (t) ≤ a 2 n t
and then

|∇ 2r y p Z (t, x, y)| = |∇ 2r y p M (σ) (F (-1) (t), x, y)| ≤ C a -(d+r) n+1
t (d+r)/2 e -c|x-y| 2 /t . We prove as in Lemma 2.6.2 that

∥Z x,n t -Z x t,u ∥ 1 ≤ C(a(u) -a n+1 )t 1/2
and then using Theorem 8.2.7 we get

d TV (Z x,n t , Z x t,u ) ≤ Ca -(d+r) n+1
(a(u) -a n+1 ) 2r/(2r+1) .

• Conclusion: considering (3.3.9), we get

d TV (X x,n t , Y x t,u ) ≤ C(1 + |x| 2 )t + Ce Ca -1 n+1 (1+|x| 2 ) t 1/2 + Ca -(d+r) n+1
(a(u) -a n+1 ) 2r/(2r+1) .

Remark 3.3.4. As in Theorem 8.2.5, we could improve the dependency in |x| in (3.3.3) and (3.3.8), at the expanse of further assumptions on V . However it would require to track the dependency in the ellipticity (in a n ) in the bounds proved in [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF], which rely on Malliavin calculus. We believe that it would considerably increase the length and the technicality of the present article, while bringing no significant improvement to our final results.

Convergence of the plateau SDE X t in total variation

In this section, we prove the convergence of the plateau SDE (X t ) defined in (3.2.21).

Exponential contraction in total variation

We first show that the property of exponential contraction that holds for the L 1 -Wasserstein distance under the setting described in Section 3.2.1 (see Theorem 2.4.2) also holds for the total variation distance.

3.4. Convergence of the plateau SDE X t in total variation Theorem 3.4.1. Let X be the solution to

X x 0 = x, dX x t = b a (X x t )dt + aσ(X x t )dW t , (3.4.1)
with a ∈ (0, A] and where b a is defined in (3.2.6), so that ν a defined in (3.2.3) is the unique invariant distribution of X ([PP23, Proposition 2.6]). Let t 0 ∈ (0, 1]. Under the assumption (3.2.12, H cf ), (a) For every x, y ∈ R d and for every t ≥ t 0 we have

d TV (X x t , X y t ) ≤ Ca -1 e C 1 /a 2 e -ρat |x -y|, ρ a := e -C 2 /a 2 . (3.4.2) (b)
For every x ∈ R d and for every t ≥ t 0 we have

d TV (X x t , ν a ) ≤ Ca -1 e C 1 /a 2 e -ρat ν a (|x -•|). (3.4.3)
Proof. (a) Following Theorem 2.4.2, we have

∀x, y ∈ R d , W 1 (X x t , X y t ) ≤ Ce C 1 /a 2 |x -y|e -ρat .
Let t ≥ t 0 and let f :

R d → R a Borel bounded function. Then E[f (X x t )] -E[f (X y t )] = E[P X t 0 f (X x t-t 0 )] -E[P X t 0 f (X x t-t 0 )],
where P X denotes the kernel associated to X. But using [PP23, Proposition 3.1] we have for every z 1 and z 2 ∈ R d ,

P X t 0 f (z 2 ) -P X t 0 f (z 1 ) = ⟨∇P X t 0 f (ξ), z 2 -z 1 ⟩ = 1 t 0 E f (X ξ t ) t 0 0 (a -1 σ -1 (X ξ s )Y ξ s ) ⊤ dW s , z 2 -z 1 ,
where ξ ∈ (z 1 , z 2 ) and (Y ξ s ) s≥0 denotes the tangent process of (X ξ s ), i.e.

Y ξ 0 = I d , dY ξ s = ∇b a (X ξ s )Y ξ s ds + a∇σ(X ξ s )Y ξ s ⊗ dW s . (3.4.4)
Since ∇σ and ∇b a are bounded (uniformly in a), we have

sup ξ∈R d ,s∈[0,t 0 ] E∥Y ξ s ∥ 2 2 < +∞,
where the bound does not depend on a. So that

P X t 0 f (z 2 ) -P X t 0 f (z 1 ) ≤ C||f || ∞ |z 2 -z 2 |a -1 sup ξ∈R d ,s∈[0,t 0 ] E∥Y ξ s ∥ 2 ,
and then [P

X t 0 f ] Lip ≤ Ca -1 ∥f ∥ ∞ . Then we obtain d TV (X x t , X y t ) ≤ Ca -1 W 1 (X x t-t 0 , X y t-t 0 ) ≤ Ca -1 e C 1 /a 2 e -ρat |x -y|.
(b) As ν a is the invariant distribution of the diffusion (3.4.1) we have

d TV (X x t , ν a ) ≤ R d d TV (X x t , X y t ) ν a (dy) ≤ Ce C 1 /a 2 e -ρat R d |x -y|ν a (dy) ≤ Ce C 1 /a 2 e -ρat ν a (|x -•|).

Convergence of the plateau SDE

Let (T n ) be the time schedule defined in (3.2.22) and by a slight abuse of notation we define

a n := a(T n ) = A log(T n + e)
and ρ n := ρ an = e -C 2 /a 2 n .

(3.4.5)

We recal Lemma 2.4.3:

0 ≤ a n -a n+1 ≍ (n log 3/2 (n)) -1 . (3.4.6) Proposition 3.4.2. Let ν a , a ∈ (0, A], be the Gibbs measure defined in (3.2.3). Assume that V is coercive, that (x → |x| 2 e -2V (x)/A 2 ) ∈ L 1 (R d ) and (3.2.2, H V 1 ). Then for n ≥ 2, d TV (ν an , ν a n+1 ) ≤ C n log(n) . (3.4.7)
Moreover, for every s, t ∈ [a n+1 , a n ], we have

d TV (ν s , ν t ) ≤ C n log(n) . (3.4.8)
The proof is given in the Appendix 3.8.1. We now prove the convergence of the SDE "by plateaux" for the total variation distance.

Theorem 3.4.3. Let X be the process defined in (3.2.21) and (3.2.23). Let t 0 be defined as in Theorem 3.4.1.

If A > max( (1 + β -1 )C 2 , (1 + β)C 1 ) where C 1 and C 2 are defined in (3.4.2), then for all x 0 ∈ R d and for all C ′ (T ) < C (T ) , for all large enough n ≥ n(C ′ (T ) ), on the time schedule (T n ) we have d TV (X x 0 Tn , ν an ) ≤ Ca -1 n n -1+(β+1)C 1 /A 2 exp -(C ′ (T ) ) 1-C 2 /A 2 (β + 1)n β-(β+1)C 2 /A 2 (1 + |x 0 |) (3.4.9) and for every t ∈ R + \ ( n≥1 [T n , T n + t 0 ]) we have d TV (X x 0 t , ν a(t) ) ≤ C(1 + |x 0 |) t (1+β) -1 -C 1 /A 2 log(t + e) .
(3.4.10)

Proof. For fixed x ∈ R d and using Theorem 3.4.1, we have for every bounded Borel function

f : R d → R, E[f (X x,n T n+1 -Tn )] -E[f (Z a n+1 )] ≤ Ca -1 n+1 e C 1 /a 2 n+1 e -ρa n+1 (T n+1 -Tn) ∥f ∥ ∞ E|x -Z a n+1 |,
where Z a n+1 ∼ ν a n+1 . Now integrating x with respect to the law of

X x 0 Tn yields d TV (X x 0 T n+1 , ν a n+1 ) ≤ Ca -1 n+1 e C 1 /a 2 n+1 e -ρa n+1 (T n+1 -Tn) W 1 (X x 0 Tn , ν an ) + W 1 (ν a n+1 , ν an ) ≤ C a -1 n+1 µ n+1 n log 3/2 (n) (1 + |x 0 |), µ n := e C 1 /a 2 n+1 e -ρa n+1 (T n+1 -Tn) (3.4.11)
where we used Theorem 2.5.1 and Proposition 2.4.4. We use the bound on µ n given by (2.5.5).

Then to bound d TV (X x 0 t , ν a n+1 ) for any t ∈ (T n +t 0 , T n+1 ), we apply Theorem (3.4.1) on the time interval [T n , t] which length is not smaller than t 0 and we conclude as in the proof of Theorem 2.5.1. Remark 3.4.4. The condition that t does not belong in any interval [T n , T n + t 0 ] is a technical condition which is specific to our strategy of proof. However this condition is not a problem for the convergence of Y t and Ȳt since for these two processes, the time schedule (T n ) is only a tool for the proof.

Convergence of Y t in total variation

We now consider (Y t ) as defined in (3.2.5) with extended definition (3.2.24).

Preliminary lemmas

Lemma 3.5.1. Let λ ∈ R + . There exists C > 0 such that for every n ≥ 0, u ≥ 0 and every x ∈ R d :

sup t≥0 E e λ|X x,n t | 2 ≤ Ce λ|x| 2 and sup t≥0 E e λ|Y x t,u | 2 ≤ Ce λ|x| 2 .
(3.5.1)

Sketch of proof. By Itō's Lemma, we have for k ≥ n and for t ∈ [T k -T n , T k+1 -T n ): d e λ|X x,n t | 2 = λe λ|X x,n t | 2 (2⟨X x,n t , dX x,n t ⟩ + d⟨X x,n ⟩ t ) + 2λ 2 |X x,n t | 2 e λ|X x,n t | 2 d⟨X x,n ⟩ t = λe λ|X x,n t | 2 -2⟨σσ ⊤ ∇V (X x,n t ), X x,n t ⟩dt + 2a 2 n+1 ⟨X x,n t , Υ(X x,n t )⟩dt + 2a n+1 ⟨X x,n t , σ(X x,n t )dW t ⟩ + a 2 n+1 Tr(σσ ⊤ (X t ))dt + 2λ 2 e λ|X x,n t | 2 a 2 n+1 (X x,n t ) ⊤ σσ ⊤ (X x,n t )X x,n t dt the "dominating" term is -⟨σσ ⊤ ∇V (X x,n t ), X x,n t ⟩dt which makes E[e λ|X x,n t | 2 ] decrease. Using assumption (3.2.12, H cf ), we have for |X x,n t | large enough, -⟨σσ ⊤ ∇V (X x,n t ), X x,n t ⟩ ≤ -C σ2 0 α 0 |X x,n t | 2 .
Moreover, using the facts that Υ and σ are bounded, that a n → 0, that |∇V | ≤ CV 1/2 and that σσ ⊤ ≥ σ2 0 I d , for large enough |X x,n t | and large enough n, the coefficient in dt in the last equation is negative. We deal with the cases where |X x,n t | is not large enough or where n is not large enough the same way as in the proof of Lemma 2.6.1 and Lemma 2.7.1, where more details can be found.

The proof is the same for Y , replacing a k+1 by a(u + t).

Proposition 3.5.2. Let T , γ > 0. There exists C > 0 such that for every Borel bounded function f : R d → R and every t ∈ (0, T ], for all n ≥ 0, for all γ < γ such that u

∈ [T n , T n+1 ] and u + t + γ ∈ [T n , T n+1 ], E P X,n t f (Y x γ,u ) -E P X,n t f (X x,n γ ) ≤ Ca -2 n+1 (a n -a n+1 )∥f ∥ ∞ γt -1 V (x). (3.5.2)
Proof. We apply Proposition 2.6.4 to g t := P X,n t f with t > 0. Following [PP23, Proposition 3.2(b)], we have

Φ gt (x) ≤ C∥f ∥ ∞ a -2 n+1 t -1 max   V 1/2 (x), sup ξ∈(X x,n γ ,Y x γ,u ) V 1/2 (ξ) 2 , V 1/2 (x) sup ξ∈(x,X x,n γ ) V 1/2 (ξ) 2   .
We conclude as in the proof of Proposition 2.6.5.

Proof of Theorem 3.2.1(a)

More precisely, we prove that for all β > 0, if

A > max (β + 1)(2C 1 + C 2 ), (1 + β -1 )C 2 , (3.5.3) then d TV Y x 0 t , ν a(t) ≤ Ce C √ log(t)(1+|x 0 | 2 ) t (1+β) -1 -(2C 1 +C 2 )/A 2 . (3.5.4)
Proof. We follow the proof of Theorem 2.2.1(b) in Section 2.7.3 based on a domino strategy with respect to some decreasing step sequence (γ n ), even though Y is not an Euler-Maruyama scheme. In this case, the step sequence (γ n ) is only a tool for the proof. This way we can choose freely the sequence (γ n ) in this section. We use Theorem 3.4.1 in place of Theorem 2.4.2 and Proposition 3.5.2 in place of Proposition 2.7.4. For f : R d → R bounded measurable and for x ∈ R d we write

Ef (X x,n T n+1 -Tn ) -Ef (Y x T n+1 -Tn,Tn ) ≤ (P Y γ init ,Tn -P X,n γ init ) • P X,n T n+1 -Γ N (Tn)+1 f (x) + N (T n+1 -T ) k=N (Tn)+2 P Y γ init ,Tn • P Y γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Y γ k-1 ,Γ k-2 • (P Y γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x) + N (T n+1 )-1 k=N (T n+1 -T )+1 P Y γ init ,Tn • P Y γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Y γ k-1 ,Γ k-2 • (P Y γ k ,Γ k-1 -P X,n γ k ) • P X,n T n+1 -Γ k f (x) + P Y γ init ,Tn •P Y γ N (Tn)+2 ,Γ N (Tn)+1 • • • • • P Y γ N (T n+1 )-1 ,Γ N (T n+1 )-2 •(P Y γ end +γ N (T n+1 ) ,Γ N (T n+1 )-1 -P X,n γ end +γ N (T n+1 ) )f (x) =: (c init ) + (a) + (b) + (c end ),
where

γ init := Γ N (Tn)+1 -T n ≤ γ N (Tn)+1 and γ end := T n+1 -Γ N (T n+1 ) ≤ γ N (T n+1 )+1 .
Then we have

(a) ≤ Ca -3 n+1 e C 1 a -2 n+1 e -ρ n+1 T n+1 ∥f ∥ ∞ V (x)(a n -a n+1 ) N (T n+1 -T ) k=N (Tn)+2 γ k e ρ n+1 Γ k ≤ Ca -3 n+1 e C 1 a -2 n+1 ∥f ∥ ∞ (a n -a n+1 )V (x)ρ -1 n+1 .
We obtain likewise

(c init ) ≤ Ca -3 n+1 e -ρ n+1 (T n+1 -Tn) ∥f ∥ ∞ (a n -a n+1 )γ N (Tn) V (x).
Applying Proposition 3.5.2 yields

(b) ≤ Ca -2 n+1 (a n -a n+1 )∥f ∥ ∞ V (x) N (T n+1 )-1 k=N (T n+1 -T )+1 γ k T n+1 -Γ k ≤ Ca -2 n+1 (a n -a n+1 )∥f ∥ ∞ V (x) log(1/γ N (T n+1 ) ).
3.6. Convergence of the Euler-Maruyama scheme in total variation Applying Proposition 3.3.3 with r = 1 along with Lemma 3.5.1 yields

(c end ) ≤ C∥f ∥ ∞ e Ca -1 n+1 (1+|x| 2 ) γ 1/2 N (Tn) + a -(d+1) n+1 (a(T n+1 -γ N (T n+1 ) ) -a n+1 ) 2/3 . But we have a(T n+1 -γ N (T n+1 ) ) -a n+1 = a(T n+1 -γ N (T n+1 ) ) -a(T n+1 ) ≤ C da dt (T n+1 ) • γ N (T n+1 ) ≤ Cγ N (T n+1 ) T n+1 .
We now choose γ n = γ 1 n -2/3 so that γ N (Tn) ≍ n -2 and then

(c end ) ≤ Ce Ca -1 n+1 (1+|x| 2 ) n -1 .
This way we obtain for every x ∈ R d :

|Ef (X x,n T n+1 -Tn ) -Ef (Y x T n+1 -Tn,Tn )| ≤ C∥f ∥ ∞ a -3 n+1 e C 1 a -2 n+1 (a n -a n+1 )V (x)ρ -1 n+1 =:v n+1 e Ca -1 n+1 (1+|x| 2 ) .
(3.5.5)

We integrate this inequality with respect to the laws of X x 0 Tn and Ȳ x 0 Tn and obtain, temporarily setting x n := X x 0 Tn and y n := Y x 0 Tn and using Lemma 2.6.1 and Lemma 3.5.1,

d TV (X x 0 T n+1 , Y x 0 T n+1 ) ≤ d TV (X xn,n T n+1 -Tn , X yn,n T n+1 -Tn ) + d TV (X yn,n T n+1 -Tn , Y ȳn T n+1 -Tn,Tn ) ≤ Ca -1 n+1 e C 1 a -2 n+1 e -ρ n+1 (T n+1 -Tn) :=µ ′ n+1 =a -1 n+1 µ n+1 d TV (X x 0 Tn , Y x 0 Tn ) + Cv n+1 e Ca -1 n+1 (1+|x 0 | 2 ) :=w n+1
, where µ n is defined in (3.4.11). Iterating this inequality yields

d TV (X x 0 T n+1 , Y x 0 T n+1 ) ≤ C(w n+1 + µ ′ n+1 w n + • • • + µ ′ n+1 • • • µ ′ 2 w 1 ) ≤ Cw n+1 ,
where we used, since A satisfies (3.5.3), that µ ′ n = O(e -Cn η ) for some η > 0 (see (2.5.5)) and that w n is bounded as it converges to 0. Moreover using Theorem 3.4.3 we have

d TV (Y x 0 Tn , ν an ) ≤ d TV (X x 0 Tn , Y x 0 Tn ) + d TV (X x 0 Tn , ν an ) ≤ Ce C √ log(n)(1+|x 0 | 2 ) n 1-(β+1)(C 1 +C 2 )/A 2 . (3.5.6) Finally, let us bound d TV (X x 0 t , Y x 0 t ) for any t ∈ [T n , T n+1 ]. If t ∈ [T n + t 0 , T n+1
] then we can apply Theorem 3.4.1 and we proceed as in the end of Section 2.6.3. If t ∈ [T n , T n + t 0 ], then we consider another shifted time schedule Tn :

= C (T ) n 1+β + 2t 0 such that ∞ i=0 [T n , T n + t 0 ] ∩ ∞ i=0 [ Tn , Tn + t 0 ] = ∅.
Making use of the new time schedule we obtain as before a bound on d TV (Y x 0 t , ν a(t) ) for every t / ∈ ∞ i=0 [ Tn , Tn + t 0 ]. Since the time schedules (T n ) and ( Tn ) are only tools for the proof of convergence of Y t , we then obtain a bound on d TV (Y t , ν a(t) ) for every t ∈ R + .

Convergence of the Euler-Maruyama scheme in total variation

We now consider ( Ȳn ) as in (3.2.17) with extended definition (3.2.25).

Preliminary lemmas

Lemma 3.6.1. Let λ ∈ R + . There exists a constant C > 0 such that for every k ≥ 0, for every u ∈ [Γ k , Γ k+1 ) and for every x ∈ R d :

sup n≥k+1 E e λ|Y x Γn-u,u | 2 ≤ Ce λ|x| 2 . (3.6.1)
Proof. The prove is the same as for Lemma 3.5.1. For the adaptation to discrete time, we refer to the proof of Lemma 2.7.1.

Proposition 3.6.2. Let T > 0. There exists C > 0 such that for every Lipschitz continuous function f and every t ∈ (0, T ], for all n ≥ 0, for all γ such that

Γ k ∈ [T n , T n+1 ], γ ≤ γ k+1 and Γ k + t + γ ∈ [T n , T n+1 ], E P t f ( Ȳ x γ,Γ k ) -E P t f (X x,n γ ) ≤ C∥f ∥ ∞ V 2 (x) a -2 n+1 t -1 γ 2 +(a(Γ k )-a n+1 )γ + a -3 n+1 t -3/2 γ 2 +γ 3/2 (a(Γ k ) -a n+1 ) . (3.6.2)
Proof. The proof is the same as the proof of Proposition 3.5.2, using Proposition 2.7.3. We also remark that we can directly improve the bound in (a n -a n+1 ) into (a(Γ k ) -a n+1 ).

Proof of Theorem 3.2.1(b)

More precisely, we prove that for all β > 0, if σ ∈ C 2r b and if

A > max (β + 1)(2C 1 + C 2 ), (1 + β -1 )C 2 (3.6.3)
and if A is large enough so that

n (β+1)C 1 /A 2 γ r/(2r+1) N (Tn) -→ n→∞ 0, (3.6.4) then d TV ( Ȳ x 0 t , ν a(t) ) ≤ C log 1/2 (t) max V 2 (x 0 ), 1 + |x 0 | t (β+1) -1 -(2C 1 +C 2 )/A 2 + e C √ log(t)(1+|x 0 | 2 ) t C 1 /A 2 γ r/(2r+1) Ct .
(3.6.5)

Proof. We still follow the proof of Theorem 2.2.1 in Section 2.7.3 based on a domino strategy, using Theorem 3.4.1 in place of Theorem 2.4.2 and Proposition 3.6.2 in place of Proposition 2.7.4.

Let n ≥ 0, for f :

R d → R bounded measurable, we split |Ef (X x,n T n+1 -Tn ) -Ef ( Ȳ x T n+1 -Tn,Tn )| into four terms (c init ), (a), (b), (c end ).
Using Theorem 3.4.1, Lemma 2.7.1 and Proposition 3.6.2 we get as in Section 2.7.3:

(a) ≤ Ca -4 n+1 e C 1 a -2 n+1 ∥f ∥ ∞ (a n -a n+1 )V 2 (x)ρ -1 n+1 . (c init ) ≤ Ca -4 n+1 e C 1 a -2 n+1 e -ρn(T n+1 -Tn) ∥f ∥ ∞ (a n -a n+1 )γ N (Tn)+1 V 2 (x).
Using Proposition 3.6.2 and Lemma 2.7.1, we obtain

(b) ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) ∥f ∥ ∞ V 2 (x) N (T n+1 )-1 k=N (T n+1 -T )+1 γ k (T n+1 -Γ k ) 3/2 3.7. Experiments + Ca -2 n+1   N (T n+1 )-1 k=N (T n+1 -T )+1 γ N (T n+1 -T ) γ k T n+1 -Γ k + N (T n+1 )-1 k=N (T n+1 -T )+1 γ k (a(Γ k )-a n+1 ) T n+1 -Γ k   ∥f ∥ ∞ V 2 (x).
But we remark that

a(Γ k ) -a n+1 = a(Γ k ) -a(T n+1 ) ≤ C da dt (T n+1 ) • (Γ k -T n+1 ) ≤ C(Γ k -T n+1 ) T n+1 log 3/2 (T n+1 )
and then

(b) ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) ∥f ∥ ∞ V 2 (x) T n+1 -γ N (T n+1 ) T n+1 -T du (T n+1 -u) 3/2 + Ca -2 n+1 γ N (T n+1 -T ) T n+1 -γ N (T n+1 ) T n+1 -T du T n+1 -u + 1 T n+1 T n+1 -γ N (T n+1 ) T n+1 -T du ∥f ∥ ∞ V 2 (x) ≤ Ca -3 n+1 γ N (T n+1 -T ) + √ γ N (T n+1 -T ) (a n -a n+1 ) ∥f ∥ ∞ V 2 (x)γ -1/2 N (T n+1 ) + Ca -2 n+1 γ N (T n+1 ) | log(γ N (T n+1 ) )| + T -1 n+1 ∥f ∥ ∞ V 2 (x) ≤ Ca -3 n+1 γ 1/2 N (T n+1 ) + (a n -a n+1 ) ∥f ∥ ∞ V 2 (x).
Applying Proposition 3.3.2 along with Lemma 3.6.1 yields

(c end ) ≤ C∥f ∥ ∞ e Ca -1 n+1 (1+|x| 2 ) γ r/(2r+1) N (T n+1 ) + a -2 n (a n -a n+1 ) .
We finally obtain for every x ∈ R d :

|Ef (X x,n T n+1 -Tn )-Ef ( Ȳ x T n+1 -Tn,Tn )| ≤ C∥f ∥ ∞ a -4 n+1 e C 1 a -2 n+1 (a n -a n+1 )V 2 (x)ρ -1 n+1 +e Ca -1 n+1 (1+|x| 2 ) γ r/(2r+1) N (T n+1 ) .
The same way as in Section 3.5.2 we get

d TV ( Ȳ x 0 T n+1 , ν a n+1 ) ≤ C a -4 n+1 e C 1 a -2 n+1 (a n -a n+1 ) max V 2 (x 0 ), 1 + |x 0 | ρ -1 n+1 + e Ca -1 n+1 (1+|x 0 | 2 ) γ r/(2r+1) N (T n+1 )
and, for t

∈ [T n , T n+1 ], d TV ( Ȳ x 0 t , ν a(t) ) ≤ Ce C 1 a -2 n+1 a -4 n+1 e C 1 a -2 n+1 (a n -a n+1 ) max V 2 (x 0 ), 1 + |x 0 | ρ -1 n+1 +e Ca -1 n+1 (1+|x 0 | 2 ) γ r/(2r+1) N (T n+1 ) .

Experiments

In this section, we compare the performances of adaptive Langevin-Simulated Annealing algorithms versus vanilla SGLD, that is the Langevin algorithm with constant (additive) σ1 . We train an artificial neural network on the CIFAR-10 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF], which is composed of RGB images of size 32 × 32 belonging to ten different classes: airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks. 50000 images are used for training and 10000 images are used for test. We use the architecture ResNet20 [START_REF] He | Deep residual learning for image recognition[END_REF] with initial width parameter 8. We consider the Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] and the RMSprop [START_REF] Tieleman | Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude[END_REF] preconditioners for the adaptive Langevin algorithms, giving L-Adam and L-RMSprop respectively. The results are given in Figure 3.1.

As pointed out in the literature [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF], the preconditioned Langevin algorithms show significant improvement compared with the vanilla SGLD algorithm. The convergence is faster and they achieve a lower error on the test set. We also display the value of the loss function on the train set during the training to show that the better performances of the preconditioned algorithms are not due to some overfitting effect.

Appendix

Proof of Proposition 3.4.2

Proof. We use the characterization of the total variation distance as the L 1 -distance between the densities, which reads

d TV (ν an , ν a n+1 ) = R d Z an e -2(V (x)-V ⋆ )/a 2 n -Z a n+1 e -2(V (x)-V ⋆ )/a 2 n+1 dx ≤ Z a n+1 R d e -2(V (x)-V ⋆ )/a 2 n -e -2(V (x)-V ⋆ )/a 2 n+1 dx + |Z an -Z a n+1 | R d e -2(V (x)-V ⋆ )/a 2 n dx = Z a n+1 a d n+1 R d e -2(V (a n+1 x)-V ⋆ )/a 2 n -e -2(V (a n+1 x)-V ⋆ )/a 2 n+1 dx + 1 - Z an Z a n+1 Z a n+1 a d n R d e -2(V (anx)-V ⋆ )/a 2 n dx.
Using (2.12.3) and (2.12.5), the first term is bounded by

C a n -a n+1 a n R d e -2(V (a n+1 y)-V ⋆ )/a 2 n V (a n+1 y) -V ⋆ a 2 n dx ≤ C a n -a n+1 a n ,

Introduction

Gibbs measures and their convergence properties are often used in stochastic optimization to minimize a function defined on R d . That is, let f : R d → R be a measurable function and let x ⋆ ∈ R d be such that f admits a global minimum at x ⋆ . It is well known [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF] that under standard assumptions, the associated Gibbs measure with density proportional to e -f (x)/t for t > 0, converges weakly to the Dirac mass at x ⋆ , δ x ⋆ , when t → 0. The Langevin equation dX s = -∇f (X s )ds + σdW s consists in a gradient descent with Gaussian noise. For σ = √ 2t, its invariant measure π t has a density proportional to e -f (x)/t (see for example [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF], Lemma 4.16), so for small t we can expect it to converge to argmin(f ) [START_REF] Arnak | Theoretical guarantees for approximate sampling from smooth and log-concave densities[END_REF] [START_REF] Vargas | Limit behavior of the invariant measure for langevin dynamics[END_REF]. The simulated annealing algorithm [START_REF] Van Laarhoven | Simulated annealing: theory and applications[END_REF] builds a Markov chain from the Gibbs measure where the parameter t converges to zero over the iterations. This idea is also used in [START_REF] Saul | Recursive stochastic algorithms for global optimization in R d[END_REF], giving a stochastic gradient descent algorithm where the noise is gradually decreased to zero. Adding a small noise to the gradient descent allows to explore the space and to escape from traps such as local minima and saddle points which appear in non-convex optimization problems [START_REF] Lazarev | Convergence of stochastic approximation procedures in the case of several roots of a regression equation[END_REF] [DPG + 14].

Such methods have been recently brought up to light again with SGLD (Stochastic Gradient Langevin Dynamics) algorithms [WT11] [LCCC16]

, especially for Machine Learning and training of artificial neural networks, which is a high-dimensional non-convex optimization problem.

The rates of convergence of Gibbs measures have been studied in [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF], [START_REF] Chii | A generalization of Laplace's method[END_REF] and [START_REF] Athreya | Gibbs measures asymptotics[END_REF] under differentiability assumptions on f . It turns out to be of order t 1/2 as soon as the Hessian matrix ∇ 2 f (x ⋆ ) is positive definite. Furthermore, in the multiple well case i.e. if the minimum of f is attained at finitely many points x ⋆ 1 , . . ., x ⋆ m , [START_REF] Hwang | Laplace's method revisited: weak convergence of probability measures[END_REF] proves that the limit distribution is a sum of Dirac masses δ x ⋆ i with coefficients proportional to det(∇ 2 f (x ⋆ i )) -1/2 as soon as all the Hessian matrices are positive definite. If such is not the case, we can conjecture that the limit distribution is concentrated around the x ⋆ i where the degeneracy is of the highest order.

The aim of this paper is to provide a rate of convergence in this degenerate setting, i.e. when x ⋆ is still a strict global minimum but ∇ 2 f (x ⋆ ) is no longer definite, which extends the range of applications of Gibbs measure-based algorithms, especially SGLD algorithms, where positive definiteness is generally assumed in the literature. In particular, our results are useful for establishing bounds for the bias made when δ x ⋆ is approximated by π t in Langevin algorithms.

As pointed out by eminent Machine Learning researchers, among them L. Bottou and Y. LeCun, in [START_REF] Sagun | Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond[END_REF] and [SEU + 17], for some classification problems using neural networks, the Hessian of the loss function at the end of the training tends out to be extremely singular. Indeed, as the dimension of the parameter which is used to minimize the loss function is large and as the neural network can be over-parametrized, many eigenvalues of the Hessian matrix are close to zero. "Therefore, a lot of methodology that assumes non-singular Hessian cannot be applied without an appropriate modification" [START_REF] Sagun | Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond[END_REF]. However, this subject is still new in the Stochastic Optimization literature and needs more theoretical funding. Thus, studying the case of a degenerate minimum helps establishing theoretically the convergence of Langevin algorithms in these cases, as in Chapter 2, where is established the convergence of Langevinsimulated annealing algorithms in the multiplicative case i.e. the diffusion has a noise coefficient σ depending on the position and slowly decreases with time ; the case where the minimum is degenerate is then dealt with using the results of the present article. Furthermore following Section 2.2.3, it is not needed to know beforehand the degree of degeneracy of the minimum in order to establish the convergence to the minimum, however knowing the degree of degeneracy additionally gives the precise rate of convergence of such Langevin algorithms.

A general framework is given in [START_REF] Athreya | Gibbs measures asymptotics[END_REF], which provides rates of convergence based on dominated convergence. However a strong and rather technical assumption on f is needed and checking it seems, to some extent, more demanding than proving the result. To be more precise, the assumption reads as follows: there exists a function g :

R d → R with e -g ∈ L 1 (R d ) and α 1 , . . . , α d ∈ (0, +∞) such that ∀h ∈ R d , 1 t [f (x ⋆ + (t α 1 h 1 , . . . , t α d h d )) -f (x ⋆ )] -→ t→0 g(h 1 , . . . , h d ). (4.1.1)
Our objective is to give conditions on f such that (4.1.1) is fulfilled and then to elucidate the expression of g depending on f and its derivatives by studying the behaviour of f at x ⋆ in every direction. In particular, we detail the different cases that can occur. Doing so we can apply the results from [START_REF] Athreya | Gibbs measures asymptotics[END_REF] yielding the convergence rate of the corresponding Gibbs measures. The orders α 1 , . . ., α d must be chosen carefully and not too large, as the function g needs to depend on every of its variables h 1 , . . ., h d , which is a necessary condition for e -g to be integrable. We also extend our results to the multiple well case.

We generally assume f to be coercive, i.e. f (x) → +∞ as ||x|| → +∞, C 2p in a neighbourhood of x ⋆ for some p ∈ N and we assume that the minimum is polynomial strict, i.e. the function f is bounded below in a neighbourhood of x ⋆ by some non-negative polynomial function, null only at x ⋆ . Thus we can apply a multi-dimensional Taylor expansion to f at x ⋆ , where the successive derivatives of f : R d → R are seen as symmetric tensors of R d . The idea is then to consider the successive subspaces where the derivatives of f are null up to some order ; using that the Taylor expansion of f (x ⋆ + h) -f (x ⋆ ) is non-negative, some cross derivative terms are null. However a difficulty arises at orders 6 and higher, as the set where the derivatives of f are null up to some order is no longer a vector subspace in general. This difficulty is linked with Hilbert's 17 th problem [START_REF] Hilbert | Ueber die Darstellung definiter Formen als Summe von Formenquadraten[END_REF], stating that a non-negative multivariate polynomial cannot be written as the sum of squares of polynomials in general. We thus need to change the definition of the subspaces we consider. Following this, we give a recursive algorithm yielding an adapted decomposition of R d into vector subspaces and a function g satisfying (4.1.1) up to a change of basis, giving a canonical higher order nested decomposition of f at x ⋆ in degenerate cases. An interesting fact is that the case where the polynomial order of x ⋆ is 10 or higher fundamentally differs from those of orders 2, 4, 6 and 8, owing to the presence of even cross terms which may be not null. The algorithm we provide works at the orders 10 or higher only under the assumption that all such even cross terms are null. In general, it is more difficult to get a general expression of g for the orders 10 and higher. We then apply our results to [START_REF] Athreya | Gibbs measures asymptotics[END_REF], where we give conditions such that the hypotheses of [START_REF] Athreya | Gibbs measures asymptotics[END_REF], especially (4.1.1), are satisfied so as to infer rates of convergence of Gibbs measures in the degenerate case where ∇ 2 f (x ⋆ ) is not necessarily positive definite. The function g given by our algorithm is a non-negative polynomial function and non-constant in any of its variables, however it needs to be assumed to be coercive to be applied to [START_REF] Athreya | Gibbs measures asymptotics[END_REF]. We study the case where g is not coercive and give a method to deal with simple generic non-coercive cases, where our algorithm seems to be a first step to a more general procedure. However, we do not give a general method in this case.

Our results are applied to Gibbs measures but they can also be applied to more general contexts, as we give a canonical higher order nested expansion of f at a minimum, in the case where some derivatives are degenerate.

For general properties of symmetric tensors we refer to [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF]. In the framework of stochastic approximation, [FP99] Section 3.1 introduced the notion of strict polynomial local extremum and investigated their properties as higher order "noisy traps".

The paper is organized as follows. In Section 4.3, we recall convergence properties of Gibbs measures and revisit the main theorem from [START_REF] Athreya | Gibbs measures asymptotics[END_REF]. This theorem requires, as an hypothesis, to find an expansion of f at its global minimum ; we properly state this problem in Section 4.3.2 under the assumption of strict polynomial minimum. In Section 4.3.3, we state our main result for both single well and multiple well cases, as well as our algorithm. In Section 4.4, we detail the expansion of f at its minimum for each order and provide the proof. We give the general expression of the canonical higher order nested expansion at any order in Section 4.4.1, where we distinguish the orders 10 and higher from the lower ones. We then provide the proof for each order 2, 4, 6 and 8 in Sections 4.4.3, 4.4.4, 4.4.6 and 4.4.7 respectively. We need to prove that, with the exponents α 1 , . . ., α d we specify, the convergence in (4.1.1) holds ; we do so by proving that, using the non-negativity of the Taylor expansion, some cross derivative terms are zero. Because of Hilbert's 17 th problem, we need to distinguish the orders 6 and 8 from the orders 2 and 4, as emphasized in Section 4.4.5. For orders 10 and higher, such terms are not necessarily zero and must then be assumed to be zero. In Section 4.4.8, we give a counter-example if this assumption is not satisfied before proving the result. In Section 4.4.9, we prove that for every order the resulting function g is constant in none of its variables and that the convergence in (4.1.1) is uniform on every compact set. In Section 4.4.10, we study the case where the function g is not coercive and give a method to deal with the simple generic case. In Section 4.5, we prove our main theorems stated in Section 4.3.3 using the expansion of f established in Section 4.4.

In Section 4.6, we deal with a "flat" example where all the derivatives in the local minimum are zero and where we cannot apply our main theorems. In Section 4.7 we give a practical example of a function f arising from an optimization problem with non definite Hessian matrix at the minimum and we numerically compute high order derivatives.

Definitions and notations

We give a brief list of notations that are used throughout the paper.

We endow R d with its canonical basis (e 1 , . . . , e d ) and the Euclidean norm denoted by || • ||. For x ∈ R d and r > 0 we denote by B(x, r) the Euclidean ball of R d of center x and radius r. For E a vector subspace of R d , we denote by p E : R d → E the orthogonal projection on E. For a decomposition of R d into orthogonal subspaces, For v 1 , . . ., v k vectors in R d and T a tensor of order k of R d , we denote the tensor product

R d = E 1 ⊕ • • • ⊕ E p , we say that an orthogonal transformation B ∈ O d (R) is adapted to this decomposition if for all j ∈ {1, . . . , p}, ∀i ∈ {dim(E 1 ) + • • • + dim(E j-1 ) + 1, . . . , dim(E 1 ) + • • • + dim(E j )}, B • e i ∈ E j .
T • (v 1 ⊗ • • • ⊗ v k ) = i 1 ,...,i k ∈{1,...,d} T i 1 •••i k v 1 i 1 . . . v k i k . More generally, if j ≤ k and v 1 , . . . , v j are j vectors in R d , then T • (v 1 ⊗ • • • ⊗ v j
) is a tensor of order k -j such that:

T • (v 1 ⊗ • • • ⊗ v j ) i j+1 ...i k = i 1 ,...,i j ∈{1,...,d} T i 1 ...i k v 1 i 1 . . . v j i j .
For h ∈ R d , h ⊗k denotes the tensor of order k such that

h ⊗k = (h i 1 . . . h i k ) i 1 ,...,i k ∈{1,...,d} . For a function f ∈ C p R d , R , we denote ∇ k f (x) the differential of order k ≤ p of f at x, as ∇ k f (x)
is the tensor of order k defined by:

∇ k f (x) = ∂ k f (x) ∂x i 1 • • • ∂x i k i 1 ,i 2 ,...,i k ∈{1,...,d}
.

By Schwarz's theorem, this tensor is symmetric, i.e. for all permutation σ ∈ S k ,

∂ k f (x) ∂x i σ(1) • • • ∂x i σ(k) = ∂ k f (x) ∂x i 1 • • • ∂x i k .
We recall the Taylor-Young formula in any dimension, and the Newton multinomial formula.

Theorem 4.2.1 (Taylor-Young formula). Let f : R d → R be C p and let x ∈ R d . Then:

f (x + h) = h→0 p k=0 1 k! ∇ k f (x) • h ⊗k + ||h|| p o(1).

Definitions and notations

We denote by k i 1 ,...,ip the p-nomial coefficient, defined as:

k i 1 , . . . , i p = k! i 1 ! . . . i p ! . Theorem 4.2.2 (Newton multinomial formula). Let h 1 , . . . , h p ∈ R d , then (h 1 + h 2 + • • • + h p ) ⊗k = i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k k i 1 , . . . , i p h ⊗i 1 1 ⊗ • • • ⊗ h ⊗ip p . (4.2.1)
For T a tensor of order k, we say that T is non-negative (resp. positive) if

∀h ∈ R d , T • h ⊗k ≥ 0 (resp. T • h ⊗k > 0). (4.2.2)
We denote L 1 (R d ) the set of measurable functions f : R d → R that are integrable with respect to the Lebesgue measure on R d . We denote by λ d the Lebesgue measure on R d . For f :

R d → R such that e -f ∈ L 1 (R d ), we define for t > 0, C t := R d e -f /t -1
and π t the Gibbs measure

π t (x)dx := C t e -f (x)/t dx.
For a family of random variables (Y t ) t∈(0,1] and Y a random variable, we write

Y t L -→ t→0 Y meaning that (Y t ) weakly converges to Y .
We give the following definition of a strict polynomial local minimum of f : Definition 4.2.3. Let f : R d → R be C 2p for p ∈ N and let x ⋆ be a local minimum of f . We say that f has a strict polynomial local minimum at x ⋆ of order 2p if p is the smallest integer such that:

∃r > 0, ∀h ∈ B(0, r) \ {0}, 2p k=2 1 k! ∇ k f (x ⋆ ) • h ⊗k > 0.
(4.2.3)

Remarks :

1. A local minimum x ⋆ of f is not necessarily strictly polynomial, for example, f : x → e -||x|| -2 and x ⋆ = 0.

2. If x ⋆ is polynomial strict, then the order is necessarily even, because if x ⋆ is not polynomial strict of order 2l for some l ∈ N, then we have h n → 0 such that the Taylor expansion in h n up to order 2l is zero ; by the minimum condition, the Taylor expansion in h n up to order 2l + 1 must be non-negative, so we also have

∇ 2l+1 f (x ⋆ ) • h ⊗2l+1 n = 0.
For f : R d → R such that min R d (f ) exists, we denote by argmin(f ) the arguments of the minima of f , i.e.

argmin(f ) = x ∈ R d : f (x) = min R d (f ) .
Without ambiguity, we write "minimum" or "local minimum" to designate f (x ⋆ ) as well as x ⋆ . Finally, we define, for x ⋆ ∈ R d and p ∈ N: R) : f admits a strict polynomial local minimum at x ⋆ of order 2p .

A p (x ⋆ ) := f ∈ C 2p (R d , R) : f admits a local minimum at x ⋆ . A ⋆ p (x ⋆ ) := f ∈ C 2p (R d ,

Convergence of Gibbs measures

Properties of Gibbs measures

Let us consider a Borel function f : R d → R with e -f ∈ L 1 (R d ). We study the asymptotic behaviour of the probability measures of density for t ∈ (0, ∞):

π t (x)dx = C t e -f (x) t dx
when t → 0. When t is small, the measure π t tends to the set argmin(f ). The following proposition makes this statement precise.

Proposition 4.3.1. Let f : R d → R be a Borel function such that

f ⋆ := essinf(f ) = inf{y : λ d {f ≤ y} > 0} > -∞,
and e -f ∈ L 1 (R d ). Then ∀ε > 0, π t ({f ≥ f ⋆ + ε}) -→ t→0 0.
Proof. As f ⋆ > -∞, we may assume without loss of generality that f ⋆ = 0 by replacing f by f -f ⋆ . Let ε > 0. It follows from the assumptions that f ≥ 0 λ d -a.e. and λ d {f ≤ ε} > 0 for every ε > 0. As e -f ∈ L 1 (R d ), we have

λ d {f ≤ ε/3} ≤ e ε/3 R d e -f dλ d < +∞.
Moreover by dominated convergence, it is clear that

C -1 t ↓ λ d {f = 0} < +∞.
We have

C t ≤ f ≤ε/3 e -f (x) t dx -1 ≤     e -ε 3t λ d {f ≤ ε 3 } >0     -1 . Then π t {f ≥ ε} = C t f ≥ε e -f (x) t dx ≤ e ε/3t f ≥ε e -f (x)/t dx λ d {f ≤ ε 3 } ≤ e -ε/3t C -1 3t λ d {f ≤ ε 3 } -→ t→0 0, because if f (x) ≥ ε, then e -f (x) t ≤ e -2ε 3t e -f (x) 3t
, and where we used that

C -1 3t ≤ C -1 1 if t ≤ 1/3 Now, let us assume that f : R d → R is continuous, e -f ∈ L 1 (R d
) and f admits a unique global minimum at x ⋆ so that argmin(f ) = {x ⋆ }. In [START_REF] Athreya | Gibbs measures asymptotics[END_REF] is proved the weak convergence of π t to δ x ⋆ and a rate of convergence depending on the behaviour of f (x ⋆ + h) -f (x ⋆ ) for small enough h. Let us recall this result in detail. 1. e -f ∈ L 1 (R d ).

For all

δ > 0, inf{f (x) -f (x ⋆ ), ||x -x ⋆ || > δ} > 0. 3. There exist α 1 , . . . , α d > 0 such that for all (h 1 , . . . , h d ) ∈ R d , 1 t [f (x ⋆ + (t α 1 h 1 , . . . , t α d h d )) -f ⋆ ] -→ t→0 g(h 1 , . . . , h d ) ∈ R. 4. R d sup 0<t<1 e -f (x ⋆ +(t α 1 h 1 ,...,t α d h d ))-f (x ⋆ ) t dh 1 . . . dh d < ∞.
For 0 < t < 1, let X t be a random vector with distribution π t . Then e -g ∈ L 1 (R d ) and

(X t -x ⋆ ) 1 t α 1 , . . . , (X t -x ⋆ ) d t α d L -→ X as t → 0 (4.3.1)
where the distribution of X has a density proportional to e -g(x 1 ,...,x d ) .

Remark: Hypothesis 2. is verified as soon as f is continuous, coercive (i.e. f (x) -→ +∞ when ||x|| → +∞) and that argmin(f ) = {0}.

To study the rate of convergence of the measure π t when t → 0 using Theorem 4.3.2, we need to identify α 1 , . . . , α d and g such that the condition (4.3.1) holds, up to a possible change of basis. Since x ⋆ is a local minimum, the Hessian

∇ 2 f (x ⋆ ) is positive semi-definite. Moreover, if ∇ 2 f (x ⋆ ) is positive definite, then choosing α 1 = • • • = α d = 1
2 , we have:

1 t [f (x ⋆ + t 1/2 h) -f (x ⋆ )] -→ t→0 1 2 h T • ∇ 2 f (x ⋆ ) • h := g(x).
And using an orthogonal change of variable:

R d e -g(x) dx = R d e -1 2 d i=1 β i y 2 i dy 1 . . . dy d < ∞,
where the eigenvalues β i are positive. However, if ∇ 2 f (x ⋆ ) is not positive definite, then some of the β i are zero and the integral does not converge.

Statement of the problem

We still consider the function f : R d → R and assume that f ∈ A ⋆ p (x ⋆ ) for some x ⋆ ∈ R d and some integer p ≥ 1. Then our objective is to find

α 1 ≥ • • • ≥ α d ∈ (0, +∞) and an orthogonal transformation B ∈ O d (R) such that: ∀h ∈ R d , 1 t [f (x ⋆ + B • (t α * h)) -f (x ⋆ )] -→ t→0 g(h 1 , . . . , h d ), (4.3.2)
where t α denotes the vector (t α 1 , . . . , t α d ) and where g : R d → R is a measurable function which is not constant in any h 1 , . . . , h d , i.e. for all i ∈ {1, . . . , d}, there exist h 1 , . . . ,

h i-1 , h i+1 , . . . , h d ∈ R d such that h i → g(h 1 , . . . , h d ) is not constant. (4.3.3)
Then we say that α 1 , . . . , α d , B and g are a solution of the problem (4.3.2). The hypothesis that g is not constant in any of its variables is important ; otherwise, we could simply take

α 1 = • • • = α d = 1
and obtain, by the first order condition:

1 t [f (x ⋆ + t(h 1 , . . . , h d )) -f (x ⋆ )] -→ t→0 0.

Main results : rate of convergence of Gibbs measures

Single well case

Theorem 4.3.3 (Single well case). Let f : R d → R be C 2p with p ∈ N and let x ⋆ ∈ R d such that:

1. f is coercive, i.e. f (x) → +∞ when ||x|| → +∞. 2. argmin(f ) = x ⋆ . 3. f ∈ A ⋆ p (x ⋆ ). 4. e -f ∈ L 1 (R d ).
Let (E k ) k , (α i ) i , B and g to be defined as in Algorithm 1 stated right after, so that for all h ∈ R d ,

1 t [f (x ⋆ + B • (t α * h)) -f (x ⋆ )] -→ t→0 g(h),
and where g is not constant in any of its variables. Moreover, assume that g is coercive and the following technical hypothesis if p ≥ 5:

∀h ∈ R d , ∀(i 1 , . . . , i p ) ∈ {0, 2, • • • , 2p} p , (4.3.4) i 1 2 + • • • + i p 2p < 1 =⇒ ∇ i 1 +•••+ip f (x ⋆ ) • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip = 0.
Then the conclusion of Theorem 4.3.2 holds, with:

1 t α 1 , . . . , 1 t α d * (B -1 • (X t -x ⋆ )) L -→ X as t → 0,
where X has a density proportional to e -g(x) .

Algorithm 1. Let f ∈ A ⋆ p (x ⋆ ) for p ∈ N. 1. Define (F k ) 0≤k≤p-1 recursively as:

F 0 = R d F k = {h ∈ F k-1 : ∀h ′ ∈ F k-1 , ∇ 2k f (x ⋆ ) • h ⊗ h ′⊗2k-1 = 0}. 2. For 1 ≤ k ≤ p -1, define the subspace E k as the orthogonal complement of F k in F k-1 .
By abuse of notation, define E p := F p-1 .

3. Define B ∈ O d (R) as an orthogonal transformation adapted to the decomposition

R d = E 1 ⊕ • • • ⊕ E p . 4. Define for 1 ≤ i ≤ d, α i := 1 2j for i ∈ {dim(E 1 ) + • • • + dim(E j-1 ) + 1, . . . , dim(E 1 ) + • • • + dim(E j )}. (4.3.5) 5. Define g : R d → R as g(h) = 2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =1 k i 1 , . . . , i p ∇ k f (x ⋆ )•p E 1 (B •h) ⊗i 1 ⊗• • •⊗p Ep (B •h) ⊗ip . (4.3.6)
Remarks :

1. The function g is not unique, as we can choose any base B adapted to the decomposition

R d = E 1 ⊕ • • • ⊕ E p .
2. The case p ≥ 5 is fundamentally different from the case p ≤ 4, since Algorithm 1 may fail to provide such (E k ) k , (α i ) i , B and g if the technical hypothesis (4.3.4) is not fulfilled, as explained in Section 4.4.8. This yields fewer results for the case p ≥ 5.

3. For p ∈ {1, 2, 3, 4}, the detail the expression of g in (4.4.7), (4.4.8), (4.4.10) and (4.4.13) respectively.

4. The function g has the following general properties : g is a non-negative polynomial of order 2p; g(0) = 0 and ∇g(0) = 0.

5. The condition on g to be coercive may seem not natural. We give more details about the case where g is not coercive in Section 4.4.10 and give a way to deal with the simple generic case of non-coercivity. However dealing with the general case where g is not coercive goes beyond the scope of our work.

6. The hypothesis that g is coercive is a necessary condition for e -g ∈ L 1 (R d ). We actually prove in Proposition 4.4.6 that it is a sufficient condition.

Practical aspect: How to check the technical hypothesis (4.3.4): Let us define

T k := ∇ k f (x ⋆ ) for k ≤ 2p and I p := {(i 1 , . . . , i p ) ∈ {0, 2, . . . , 2p} 2p : i 1 /2 + • • • + i p /(2p) < 1}.
We first apply Algorithm 1. We perform the change of basis given by B; then the new derivative tensor after change of coordinates is given by (B ⊤ ) ⊗k • T k . To simplify the notations, let us assume that B = I d . For every (i 1 , . . . , i p ) ∈ I p we check the condition (4.3.4) as follows.

Let

k := i 1 + • • • + i p . The function h → T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip
is a polynomial function, so we need to check that all of its coefficient are null. Let us define the sets of indexes:

I ℓ := {i 1 + • • • + i ℓ-1 + 1, . . . , i 1 + • • • + i ℓ }, 1 ≤ ℓ ≤ p, L ℓ := {dim(E 1 ) + • • • + dim(E ℓ-1 ) + 1, . . . , dim(E 1 ) + • • • + dim(E ℓ )}, 1 ≤ ℓ ≤ p.
Having in mind that B = I d so that for all h ∈ R d , 1 ≤ m ≤ p and 1 ≤ j ≤ d, we have [p Em (h)] j = h j if j ∈ L m and 0 otherwise. Then we have

T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip = j 1 ,...,j k ∈{1,...,d} [T k ] j 1 ,...,j k p m=1 ℓ∈Im [p Em (h)] j l = (j 1 ,...,j k )∈L i 1 1 ו••×L ip p [T k ] j 1 ,...,j k k ℓ=1 h j ℓ .
Moreover, having in mind that the tensor T k is a symmetric tensor, we deduce that the condition (4.3.4) is satisfied if and only if

∀(i 1 , . . . , i p ) ∈ I p , ∀(j 1 , . . . , j i 1 +•••+ip ) ∈ L i 1 1 × • • • × L ip p , [T i 1 +•••+ip ] j 1 ,...,j i 1 +•••+ip = 0.
Practical aspect: If the function f is analytically known, then the derivative tensors can be computed analytically, in particular using automatic differentiation. If the function f is not analytically known, in particular if f is known only at some points, then the present article gives possible convergence rates of Gibbs-based minimization algorithms and proves that the convergence rate is still polynomial under the assumption that the minimum is strictly polynomial. Degenerate cases yield a different behaviour that can be empirically detected; the order of degeneracy can be estimated as well.

Multiple well case

Still following [START_REF] Athreya | Gibbs measures asymptotics[END_REF], we study the multiple well case, i.e. the global minimum is attained in a finite number of points in R d , say {x ⋆ 1 , . . . , x ⋆ m } for some m ∈ N. In this case, the limiting measure of π t will have its support in {x ⋆ 1 , . . . , x ⋆ m }, with different weights.

Theorem 4.3.4 [START_REF] Athreya | Gibbs measures asymptotics[END_REF].

Let f : R d → R be a Borel function, let f ⋆ ∈ R and let x ⋆ 1 , . . . , x ⋆ m ∈ R d such that : 1. e -f ∈ L 1 (R d ).

For all

δ > 0, inf{f (x) -f ⋆ , ||x -x ⋆ i || > δ, 1 ≤ i ≤ m} > 0.
3. There exist

(α ij ) 1≤i≤m 1≤j≤d
such that for all i, j, α ij ≥ 0 and for all i:

1 t [f (x ⋆ i + (t α i1 h 1 , . . . , t α id h d )) -f ⋆ ] -→ t→0 g i (h 1 , . . . , h d ) ∈ [0, ∞).
4. For all i ∈ {1, . . . , m},

R d sup 0<t<1 e - f (x ⋆ i +(t α i1 h 1 ,...,t α id h d ))-f ⋆ t dh 1 . . . dh d < ∞.
Then, let α := min 1≤i≤m d j=1 α ij and let J := i ∈ {1, . . . , m} :

d j=1 α ij = α . For 0 < t < 1,
let X t be a random vector with distribution π t . Then:

X t L -→ t→0 1 j∈J R d e -g j (x) dx i∈J R d e -g i (x) dx • δ x ⋆ i .
Theorem 4.3.5 (Multiple well case). Let f :

R d → R be C 2p for p ∈ N, let f ⋆ ∈ R and let x ⋆ 1 , . . . , x ⋆ m ∈ R d such that: 1. f is coercive i.e. f (x) → +∞ when ||x|| → +∞. 2. argmin(f ) = {x ⋆ 1 , . . . , x ⋆ m } and for all i, f (x ⋆ i ) = f ⋆ . 3. For all i ∈ {1, . . . , m}, f ∈ A ⋆ p i (x ⋆ i ) for some p i ≤ p. 4. e -f ∈ L 1 (R d ).
Then, for every i ∈ {1, . . . , m}, we consider (E ik ) k , (α ij ) j , B i and g i as defined in Algorithm 1, where we consider f to be in A ⋆ p i (x ⋆ i ), so that for every h ∈ R d :

1 t [f (x ⋆ i + B i • (t α i * h)) -f ⋆ ] -→ t→0 g i (h 1 , . . . , h d ) ∈ [0, ∞),
where t α i is the vector (t α i1 , . . . , t α id ) and where g i is not constant in any of its variables. Furthermore, we assume that for all i, g i is coercive and the following technical hypothesis for every i such that p i ≥ 5:

∀h ∈ R d , ∀(i 1 , . . . , i p i ) ∈ {0, 2, . . . , 2p i } p i , i 1 2 + • • • + i p i 2p < 1 =⇒ ∇ i 1 +•••+ip i f (x ⋆ i ) • p E i1 (h) ⊗i 1 ⊗ • • • ⊗ p E ip i (h) ⊗ip i = 0.
Let α := min 1≤i≤m d j=1 α ij and let J := i ∈ {1, . . . , m} :

d j=1 α ij = α . Then: X t -→ t→0 1 j∈J R d e -g j (x) dx i∈J R d e -g i (x) dx • δ x ⋆ i .
Moreover, let δ > 0 be small enough so that the balls B(x ⋆ i , δ) are disjoint, and define the random vector X it to have the law of X t conditionally to the event ||X t -x ⋆ i || < δ. Then:

1 t α i1 , . . . , 1 t α id * (B -1 i • (X it -x ⋆ i )) L -→ X i as t → 0,
where X i has a density proportional to e -g i (x) .

Expansion of f at a local minimum with degenerate derivatives

In this section, we aim at answering to the problem stated in (4.3.2) in order to devise conditions to apply Theorem 4.3.2. This problem can also be considered in a more general setting, independently of the study of the convergence of Gibbs measures. It provides a non degenerate higher order nested expansion of f at a local minimum when some of the derivatives of f are degenerate. Note here that we only need x ⋆ to be a local minimum instead of a global minimum, since we only give local properties.

For k ≤ 2p, we define the tensor of order k, T k := ∇ k f (x ⋆ ).

Expansion of f for any order p

In this section, we state our result in a synthetic form. The proofs of the cases p = 1, 2, 3, 4 are individually detailled in Sections 4.4.3, 4.4.4, 4.4.6 and 4.4.7 respectively.

Theorem 4.4.1. Let f : R d → R be C 2p for some p ∈ N and assume that f ∈ A ⋆ p (x ⋆ ) for some

x ⋆ ∈ R d . 1. If p ∈ {1, 2, 3, 4}, then there exists orthogonal subspaces of R d , E 1 , . . . , E p such that R d = E 1 ⊕ • • • ⊕ E p ,
and satisfying for every h ∈ R d :

1 t f x ⋆ + t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h) -f (x ⋆ ) (4.4.1) -→ t→0 2p k=2 1 k! i 1 ,...,ip∈{0,••• ,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =1 k i 1 , . . . , i p T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip . (4.4.2)
The convergence is uniform with respect to h on every compact set. Moreover, let B ∈ O d (R) be an orthogonal transformation adapted to the decomposition

E 1 ⊕ • • • ⊕ E p , then 1 t [f (x ⋆ + B • (t α * h)) -f (x ⋆ )] -→ t→0 g(h), (4.4.3)
where

g(h) = 2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =1 k i 1 , . . . , i p T k • p E 1 (B • h) ⊗i 1 ⊗ • • • ⊗ p Ep (B • h) ⊗ip (4.4.4)
is not constant in any of its variables h 1 , . . . , h d and

α i := 1 2j for i ∈ {dim(E 1 ) + • • • + dim(E j-1 ) + 1, . . . , dim(E 1 ) + • • • + dim(E j )}.
(4.4.5)

2. If p ≥ 5 and if there exist orthogonal subspaces of R d , E 1 , . . . , E p such that

R d = E 1 ⊕ • • • ⊕ E p
and satisfying the following additional assumption

∀h ∈ R d , ∀(i 1 , . . . , i p ) ∈ {0, 2, . . . , 2p} p , (4.4.6) i 1 2 + • • • + i p 2p < 1 =⇒ T i 1 +•••+ip • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip = 0, then (4.4.
2) stills holds true, as well as the uniform convergence on every compact set. Moreover, if B ∈ O d (R) is an orthogonal transformation adapted to the previous decomposition, then (4.4.3) still hold true. However, depending on the function f , such subspaces do not necessarily exist.

Remarks:

1. The limit (4.4.2) can be rewritten as:

2p k=2 i 1 ,...,ip∈{0,••• ,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =1 T k • p E 1 (h) ⊗i 1 i 1 ! ⊗ • • • ⊗ p Ep (h) ⊗ip i p ! .
2. For p ∈ {1, 2, 3, 4}, we explicitly give the expression of the sum (4.4.2) and the p-tuples

(i 1 , . . . , i p ) such that i 1 2 + • • • + ip 2p = 1, in (4.4.7
), (4.4.8), (4.4.10) and (4.4.13) respectively. 3. For p ∈ {1, 2, 3, 4}, we give in Algorithm 1 an explicit construction of the orthogonal subspaces E 1 , . . . , E p as complementaries of annulation sets of some derivatives of f . 4. The case p ≥ 5 is fundamentally different from the case p ∈ {1, 2, 3, 4}. The strategy of proof developed for p ∈ {1, 2, 3, 4} fails if the assumption (4.4.6) is not satisfied. In 4.4.8 a counter-example is detailed. The case p ≥ 5 yields fewer results than for p ≤ 4, as the assumption (4.4.6) is strong.

5. For p ≥ 5, such subspaces E 1 , . . ., E p may also be obtained from Algorithm 1, however (4.4.6) is not necessarily true in this case.

The proof of Theorem 4.4.1 is given first individually for each p ∈ {1, 2, 3, 4}, in Sections 4.4.3, 4.4.4, 4.4.6, 4.4.7 respectively. The proof for p ≥ 5 is given in Section 4.4.8. The proof of the uniform convergence and of the fact that g is not constant is given in Section 4.4.9.

Review of the one dimensional case

We review the case d = 1, as it guides us for the proof in the case d ≥ 2. The strategy is to find the first derivative f (m) (x ⋆ ) which is non zero and then to choose α 1 = 1/m. Proposition 4.4.2. Let f : R → R be C p for some p ∈ N and let x ⋆ be a strict polynomial local minimum of f . Then :

1. The order of the local minimum m is an even number and f (m) (x ⋆ ) > 0.

f (x

⋆ + h) = h→0 f (x ⋆ ) + f (m) (x ⋆ ) m! h p + o(h m )
Then α 1 := 1/m is the solution of (4.3.2) and

1 t (f (x ⋆ + t 1/m h) -f (x ⋆ )) -→ t→0 f (m) (x ⋆ ) m! h m
which is a non-constant function of h, since f (m) (x ⋆ ) ̸ = 0. The direct proof using the Taylor formula is left to the reader.

Proof of Theorem 4.4.1 for

p = 1 Let f ∈ A ⋆ 1 (x ⋆ ).
The assumption that x ⋆ is a strict polynomial local minimum at order 2 implies that ∇ 2 f (x ⋆ ) is positive definite. Let us denote (β i ) 1≤i≤d its positive eigenvalues. By the spectral theorem, let us write

∇ 2 f (x ⋆ ) = BDiag(β 1:d )B T for some B ∈ O d (R). Then: 1 t (f (x ⋆ + t 1/2 B • h) -f (x ⋆ )) -→ t→0 1 2 d i=1 β i h 2 i . (4.4.7)
Thus, a solution of (4.3.2) is

α 1 = • • • = α d = 1 2 , B, and g(h 1 , . . . , h d ) = 1 2 d i=1 β i h 2 i
, which is a non-constant function of every h 1 , . . . , h d , since for all i, β i is positive.

In the following, our objective is to establish a similar result when ∇ 2 f (x ⋆ ) is not necessarily positive definite.

Proof of Theorem 4.4.1 for p = 2

Theorem 4.4.3. Let f ∈ A 2 (x ⋆ ). Then there exist orthogonal subspaces E and F such that R d = E ⊕ F , and that for all h ∈ R d :

1 t f (x ⋆ + t 1/2 p E (h) + t 1/4 p F (h)) -f (x ⋆ ) -→ t→0 1 2 ∇ 2 f (x ⋆ ) • p E (h) ⊗2 + 1 2 ∇ 3 f (x ⋆ ) • p E (h) ⊗ p F (h) ⊗2 + 1 4! ∇ 4 f (x ⋆ ) • p F (h) ⊗4 . (4.4.8) Moreover, if f ∈ A ⋆ 2 (x ⋆ )
, then this is a solution to the problem (4.3.2), with E 1 = E, E 2 = F , α defined in (4.4.5), B adapted to the previous decomposition and g defined in (4.4.4).

Remark:

The set of 2-tuples (i 1 , i 2 ) such that i 1 2 + i 2 4 = 1, are (2, 0), (1, 2) and (0, 4), which gives the terms appearing in the sum in (4.4.2).

Proof. Let

F := {h ∈ R d : ∇ 2 f (x ⋆ ) • h ⊗2 = 0}. By the spectral theorem and since ∇ 2 f (x ⋆ ) is positive semi-definite, F = {h ∈ R d : ∇ 2 f (x ⋆ ) • h = 0 ⊗1 } is a vector subspace of R d . Let E be the orthogonal complement of F in R d .
For h ∈ R d we expand the left term of (4.4.8) using the Taylor formula up to order 4 and the multinomial formula (4.2.1), giving

4 k=2 1 k! i 1 ,i 2 ∈{0,...,k} i 1 +i 2 =k k i 1 , i 2 t i 1 2 + i 2 4 -1 T k • p E (h) ⊗i 1 ⊗ p F (h) ⊗i 2 + o(1).
The terms with coefficient t a , a > 0, are o(1) as t → 0. By definition of F we have ∇ 2 f (x ⋆ ) • p F (h) = 0 ⊗1 , so we also have

∇ 3 f (x ⋆ ) • p F (h) ⊗3 = 0
by the local minimum condition. This yields the convergence stated in (4.4.8). Moreover, if x ⋆ is a local minimum of polynomial order 4, then by the local minimum condition, ∇ 4 f (x ⋆ ) > 0 on F in the sense of (4.2.2). Moreover, since ∇ 2 f (x ⋆ ) > 0 on E, then the limit is not constant in any h 1 , . . . , h d .

Remark:

The cross odd term is not necessarily null. For example, consider

f : R 2 -→ R (x, y) -→ x 2 + y 4 + xy 2 .
Then f admits a global minimum at x ⋆ = 0 since |xy 2 | ≤ 1 2 (x 2 + y 4 ). We have

E 1 = R(1, 0), E 2 = R(0, 1) and for all (x, y) ∈ R 2 , T 3 • (xe 1 ) ⊗ (ye 2 ) ⊗2 = 2xy 2 is not identically null.

Difficulties beyond the 4th order and Hilbert's 17 th problem

If we do not assume as in the previous section that ∇ 4 f (x ⋆ ) is not positive on F , then we carry on the development of f (x ⋆ + h) up to higher orders. A first idea is to consider F 2 := {h ∈ F : ∇ 4 f (x ⋆ ) • h ⊗4 = 0} ⊆ F and E 2 a complement subspace of F 2 in F , and to continue this process by induction as in Section 4.4.4. However, F 2 is not necessarily a subspace of F .

Indeed, let T be a symmetric tensor defined on R d ′ of order 2k with k ∈ N. As T is symmetric, there exist vectors v 1 , . . . , v q ∈ R d ′ , and scalars λ 1 , . . . , λ q ∈ R such that T = i λ i (v i ) ⊗2k (see [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF], Lemma 4.2.), so

∀h ∈ R d ′ , T • h ⊗2k = q i=1 λ i (v i ) ⊗2k • h ⊗2k = q i=1 λ i ⟨v i , h⟩ 2k . For k = 2 and T = ∇ 2k f (x ⋆ ) | F , since x ⋆ is a local minimum, we have, identifying F and R d ′ , ∀h ∈ R d ′ , T • h ⊗2k ≥ 0
Then, we could think it implies that for all i, λ i ≥ 0, and then

T • h ⊗2k = 0 =⇒ ∀i, ⟨v i , h⟩ = 0
which would give a linear caracterization of {h ∈ R d ′ : T • h ⊗2k = 0} and in this case, F 2 would be a subspace of F . However this reasoning is not correct in general as we do not have necessarily that for all i, λ i ≥ 0.

We can build counter-examples as follows. Since T is a non-negative symmetric tensor, T can be seen as a non-negative homogeneous polynomial of degree 2k with d ′ variables. A counterexample at order 2k = 4 is T (X, Y, Z) = ((X -Y )(X -Z)) 2 , which is a non-negative polynomial of order 4, but {T = 0} = {X = Y or X = Z}, which is not a vector space.

Another counterexample given in [START_REF] Motzkin | The arithmetic-geometric inequality[END_REF] at order 2k = 6 is the following. We define

T (X, Y, Z) = Z 6 + X 4 Y 2 + X 2 Y 4 -3X 2 Y 2 Z 2
By the arithmetic-geometric mean inequality and its equality case, T is non-negative and T (x, y, z) = 0 if and only if z 6 = x 4 y 2 = x 2 y 4 , so that

{T = 0} = R    1 1 1    ∪ R    -1 1 1    ∪ R    1 -1 1    ∪ R    1 1 -1    .
Hence, {T = 0} is not a subspace of R 3 . In particular T cannot be written as i λ i (v i ) ⊗2k with λ i ≥ 0.

In fact, this problem is linked with the Hilbert's seventeenth problem that we recall below.

Problem 1 (Hilbert's seventeeth problem). Let P be a non-negative polynomial with d ′ variables, homogeneous of even degree 2k. Find polynomials P 1 , . . . , P r with d ′ variables, homogeneous of degree k, such that P = r i=1 P 2 i Hilbert proved in 1888 [START_REF] Hilbert | Ueber die Darstellung definiter Formen als Summe von Formenquadraten[END_REF] that there does not always exist a solution. In general {T = 0} is not even a submanifold of R d ′ . Indeed, taking T :

h → ∇ 2k f (x ⋆ ) • h ⊗2k , we have ∂ h T • h = 2k∇ 2k f (x ⋆ ) • h ⊗2k-1 is not surjective in h = 0,
so the surjectivity condition for {T = 0} to be a submanifold is not fulfilled.

Proof of Theorem 4.4.1 for p = 3

We slightly change our strategy of proof developed in Section 4.4.4. For k ≥ 2, we define F k recursively as

F k := {h ∈ F k-1 : ∀h ′ ∈ F k-1 , ∇ 2k f (x ⋆ ) • h ⊗ h ′⊗2k-1 = 0}, (4.4.9) instead of {h ∈ F k-1 : ∇ 2k f (x ⋆ ) • h ⊗2k = 0}. Then, by construction, F k is a vector subspace of R d . Theorem 4.4.4. Let f ∈ A 3 (x ⋆ ).
Then there exist orthogonal subspaces of R d , E 1 , E 2 and F 2 , such that

R d = E 1 ⊕ E 2 ⊕ F 2 ,
and such that for all h ∈ R d ,

1 t f (x ⋆ + t 1/2 p E 1 (h) + t 1/4 p E 2 (h) + t 1/6 p F 2 (h)) -f (x ⋆ ) (4.4.10) -→ t→0 1 2 ∇ 2 f (x ⋆ ) • p E 1 (h) ⊗2 + 1 2 ∇ 3 f (x ⋆ ) • p E 1 (h) ⊗ p E 2 (h) ⊗2 + 1 4! ∇ 4 f (x ⋆ ) • p E 2 (h) ⊗4 + 4 4! ∇ 4 f (x ⋆ ) • p E 1 (h) ⊗ p F 2 (h) ⊗3 + 10 5! ∇ 5 f (x ⋆ ) • p E 2 (h) ⊗2 ⊗ p F 2 (h) ⊗3 + 1 6! ∇ 6 f (x ⋆ ) • p F 2 (h) ⊗6 . Moreover, if f ∈ A ⋆ 3 (x ⋆ )
, then this is a solution to the problem (4.3.2), with E 3 = F 2 , α defined in (4.4.5), B adapted to the previous decomposition and g defined in (4.4.4).

Remark:

The set of 3-tuples (i 1 , i 2 , i 3 ) such that i 1 2 + i 2 4 + i 3 6 = 1, are (2, 0, 0), (1, 2, 0), (0, 4, 0), (1, 0, 3), (0, 2, 3), (0, 0, 6), which gives the terms appearing in (4.4.2).

Proof. We consider the subspace

F 1 := {h ∈ R d : T 2 • h ⊗2 = 0} = {h ∈ R d : T 2 • h = 0 ⊗1 },
since T 2 ≥ 0. Then, let E 1 be the orthogonal complement of F 1 in R d and consider the vector subspace of F 1 defined by

F 2 = {h ∈ F 1 : ∀h ′ ∈ F 1 , T 4 • h ⊗ h ′⊗3 = 0}.
Let E 2 be the orthogonal complement of F 2 in F 1 . Then we have

R d = E 1 ⊕ F 1 = E 1 ⊕ E 2 ⊕ F 2 .
For h ∈ R d we expand the left term of (4.4.10) using the Taylor formula up to order 6 and the multinomial formula (4.2.1), giving

6 k=2 1 k! i 1 ,i 2 ,i 3 ∈{0,...,k} i 1 +i 2 +i 3 =k k i 1 , i 2 , i 3 t i 1 2 + i 2 4 + i 3 6 -1 T k • p E 1 (h) ⊗i 1 ⊗ p E 2 (h) ⊗i 2 ⊗ p F 2 (h) ⊗i 3 + o(1),
and we prove the convergence stated in (4.4.10).

All the terms with coefficient t a where a > 0 are o(1) as t → 0.

Order 2: we have

T 2 • p E 2 (h) = 0 ⊗1 and T 2 • p F 2 (h) = 0 ⊗1 so the only term for k = 2 is 1 2 T 2 • p E 1 (h) ⊗2 . Order 3: ▷ Since x ⋆ is a local minimum and T 2 • p F 1 (h) ⊗2 = 0, we have T 3 • p F 1 (h) ⊗3 = 0.
Then, using property Proposition 4.8.1, if the factor p E 1 (h) does not appear as an argument in T 3 , then the corresponding term is zero.

▷ Let us prove that

T 3 • p E 1 (h) ⊗ p F 2 (h) ⊗2 = 0.
Using Theorem 4.4.3 with

E = E 1 , F = E 2 ⊕ F 2 , we have in particular that for all h ∈ R d , 1 2 T 2 • p E (h) ⊗2 + 1 2 T 3 • p E (h) ⊗ p F (h) ⊗2 + 1 4! T 4 • p F (h) ⊗4 ≥ 0. (4.4.11) Then taking h ∈ E 1 ⊕ F 2 so that h = p E 1 (h) + p F 2 (h) and with T 4 • p F 2 (h) |F 1 ≡ 0 ⊗3 , (4.4.12)
we may rewrite (4.4.11) as

1 2 T 2 • p E 1 (h) ⊗2 + 1 2 T 3 • p E 1 (h) ⊗ p F 2 (h) ⊗2 ≥ 0.
Now, considering h ′ = λh, we have that for all λ ∈ R,

λ 2 1 2 T 2 • p E 1 (h) ⊗2 + λ 2 T 3 • p E 1 (h) ⊗ p F 2 (h) ⊗2 ≥ 0, so that necessarily T 3 • p E 1 (h) ⊗ p F 2 (h) ⊗2 = 0. ▷ Let us prove that T 3 • p E 1 (h) ⊗ p E 2 (h) ⊗ p F 2 (h) = 0.
We use again (4.4.11), with p

F (h) = p E 2 (h) + p F 2 (h), so that 1 2 T 2 • p E 1 (h) ⊗2 + 1 2 T 3 • p E 1 (h) ⊗ p E 2 (h) + p F 2 (h) ⊗2 + 1 4! T 4 • p E 2 (h) + p F 2 (h) ⊗4 ≥ 0.
But using (4.4.12) and that ⊗2 . Order 4: If the factor p E 1 (h) does not appear and if the factor p F 2 (h) appears at least once, then using (4.4.12) the corresponding term is zero. If p E 1 (h) appears, the only term with a nonpositive exponent of

T 3 • p E 1 (h) ⊗ p F 2 (h) ⊗2 = 0, we obtain 1 2 T 2 • p E 1 (h) ⊗2 + 1 2 T 3 • p E 1 (h)⊗p E 2 (h) ⊗2 + T 3 • p E 1 (h)⊗p E 2 (h)⊗p F 2 (h) + 1 4! T 4 • p E 2 (h) ⊗4 ≥ 0. Now, considering h ′ = p E 1 (h) + p E 2 (h) + λp F 2 (h), we have that for all λ ∈ R, 1 2 T 2 • p E 1 (h) ⊗2 + 1 2 T 3 • p E 1 (h)⊗p E 2 (h) ⊗2 + λT 3 • p E 1 (h)⊗p E 2 (h)⊗p F 2 (h) + 1 4! T 4 • p E 2 (h) ⊗4 ≥ 0, so necessarily T 3 • p E 1 (h) ⊗ p E 2 (h) ⊗ p F 2 (h) = 0. ▷ The last remaining term for k = 3 is 1 2 T 3 • p E 1 (h) ⊗ p E 2 (h)
t is 4 4! T 4 • p E 1 (h) ⊗ p F 2 (h) ⊗3 . So the only terms for k = 4 are 1 4! T 4 • p E 2 (h) ⊗4 and 4 4! T 4 • p E 1 (h) ⊗ p F 2 (h) ⊗3 .
Order 5: ▷ The terms where p E 1 (h) appears at least once have a coefficient t a with a > 0 so are o(1) when t → 0.

▷ We have T 2 • p F 2 (h) ⊗2 = 0, T 3 • p F 2 (h) ⊗3 = 0, T 4 • p F 2 (h) ⊗4 = 0 and since x ⋆ is a local minimum, we have T 5 • p F 2 (h) ⊗5 = 0. ▷ Let us prove that T 5 • p E 2 (h) ⊗ p F 2 (h) ⊗4 = 0. Let h ∈ R d . We have 1 t 11/12 f (x ⋆ + t 1/4 p E 2 (h) + t 1/6 p F 2 (h)) -f (x ⋆ ) -→ t→0 1 4! T 5 • p E 2 (h) ⊗ p F 2 (h) ⊗4 ≥ 0.
Hence, considering h ′ = λh, we have for every λ ∈ R,

λ 5 T 5 • p E 2 (h) ⊗ p F 2 (h) ⊗4 ≥ 0,
which yields the desired result. ▷ The only remaining term for p = 5 is

10 5! T 5 • p E 2 (h) ⊗2 ⊗ p F 2 (h) ⊗3 .
Order 6: The only term for k = 6 is 1 6! T 6 • p F 2 (h) ⊗6 ; the other terms have a coefficient t a with a > 0, so are o(1) when t → 0.

Remark : As in Theorem 4.4.3 and the remark that follows, the remaining odd cross-terms cannot be proved to be zero using the same method of proof, and may be actually not zero. For example, consider:

f : R 2 -→ R (x, y) -→ x 4 + y 6 + x 2 y 3 , which satisfies h → ∇ 5 f (x ⋆ ) • p E 2 (h) ⊗2 ⊗ p F 2 (h) ⊗3 ̸ ≡ 0.
Order 2

(2, 0, 0, 0) Order 3

(2, 1, 0, 0) Order 4 (0, 4, 0, 0), (1, 1, 0, 2), (1, 0, 3, 0) Order 5 (1, 0, 0, 4), (0, 2, 3, 0), (0, 3, 0, 2) Order 6 (0, 1, 3, 2), (0, 2, 0, 4), (0, 0, 6, 0) Order 7 (0, 1, 0, 6), (0, 0, 3, 4) Order 8 (0, 0, 0, 8) Table 4.1: Terms expressed as 4-tuples in the development (4.4.13)

E 1 F 1 T 2 ≥ 0 T 2 = 0 E 2 F 2 T 4 ≥ 0 T 4 = 0 E 3 F 3 T 6 ≥ 0 T 6 = 0
f ∈ A 4 (x ⋆ ). Then there exist orthogonal subspaces of R d , E 1 , E 2 , E 3 and F 3 such that R d = E 1 ⊕ E 2 ⊕ E 3 ⊕ F 3 ,
and for all h ∈ R d ,

1 t f (x ⋆ + t 1/2 p E 1 (h) + t 1/4 p E 2 (h) + t 1/6 p E 3 (h) + t 1/8 p F 3 (h)) -f (x ⋆ ) -→ t→0 8 k=2 1 k! i 1 ,...,i 4 ∈{0,...,k} i 1 +•••+i 4 =k k i 1 , . . . , i 4 T k • p E 1 (h) ⊗i 1 ⊗ p E 2 (h) ⊗i 2 ⊗ p E 3 (h) ⊗i 3 ⊗ p F 3 (h) ⊗i 4 .
(4.4.13)

These terms are summarized as tuples (i 1 , . . . , i 4 ) in Table 4.1. Moreover, if f ∈ A ⋆ 4 (x ⋆ ), then this is a solution to (4.3.2), with E 4 = F 3 , α defined in (4.4.5), B adapted to the previous decomposition and g defined in (4.4.4).

Proof. As before, we define the subspaces F 0 := R d and by induction:

F k = h ∈ F k-1 : ∀h ′ ∈ F k-1 , T 2k • h ⊗ h ′⊗3 = 0 for k = 1, 2, 3. We define E k as the orthogonal complement of F k in F k-1 for k = 1, 2, 3, so that R d = E 1 ⊕ E 2 ⊕ E 3 ⊕ F 3 .
Then we apply a Taylor expansion up to order 8 to the left side of (4.4.13) and the multinomial formula (4.2.1), which reads

8 k=2 1 k! i 1 ,...,i 4 ∈{0,...,k} i 1 +•••+i 4 =k k i 1 , . . . , i 4 t i 1 2 +•••+ i 4 8 -1 T k •p E 1 (h) ⊗i 1 ⊗p E 2 (h) ⊗i 2 ⊗p E 3 (h) ⊗i 3 ⊗p F 3 (h) ⊗i 4 +o(1). ▷ If i 1 2 + • • • + i 4 8 > 1 then the corresponding term is in o(1) when t → 0. ▷ If i 1 2 + • • • + i 4
8 < 1 then the corresponding term diverges when t → 0, so we need to prove that actually

T k • p E 1 (h) ⊗i 1 ⊗ p E 2 (h) ⊗i 2 ⊗ p E 3 (h) ⊗i 3 ⊗ p F 3 (h) ⊗i 4 = 0. (4.4.14) -If i 1 2 + i 2 4 + i 3 6 + i 4 8 < 1 but if we also have i 1 2 + i 2 4 + i 3 6 + i 4
6 < 1, then by applying the property at the order 6 (Theorem 4.4.4) with the 3-tuple (i 1 , i 2 , i 3 + i 4 ), we get (4.4.14).

-So we only need to consider 4-tuples such that i 1 2 + i 2 4 + i 3 6 + i 4 8 < 1 and i 1 2 + i 2 4 + i 3 6 + i 4 6 ≥ 1. We can remove all the terms which are null by the definitions of the subspaces E 1 , E 2 , E 3 , F 3 . The remaining terms are:

For k = 4:

t 21/24 6 T 4 • p E 1 (h) ⊗ p F 3 (h) ⊗3 , t 11/12 2 T 4 • p E 1 (h) ⊗ p E 3 (h) ⊗ p F 3 (h) ⊗2 , t 23/24 2 T 4 • p E 1 (h) ⊗ p E 3 (h) ⊗2 ⊗ p F 3 (h). For k = 5 : t 21/24 12 T 5 • p E 2 (h) ⊗2 ⊗ p F 3 (h) ⊗3 , t 11/12 4 T 5 • p E 2 (h) ⊗2 ⊗ p E 3 (h) ⊗ p F 3 (h) ⊗2 , t 23/24 4 T 5 • p E 2 (h) ⊗2 ⊗ p E 3 (h) ⊗2 ⊗ p F 3 (h). For k = 6 : t 21/24 5! T 6 •p E 2 (h)⊗p F 3 (h) ⊗5 , t 11/12 4! T 6 •p E 2 (h)⊗p E 3 (h)⊗p F 3 (h) ⊗4 , t 23/24 12 T 6 •p E 2 (h)⊗ p E 3 (h) ⊗2 ⊗ p F 3 (h) ⊗3 .
First, we note that

1 t 21/24 f (x ⋆ + t 1/2 p E 1 (h) + t 1/4 p E 2 (h) + t 1/6 p E 3 (h) + t 1/8 p F 3 (h)) -f (x ⋆ ) -→ t→0 1 6 T 4 • p E 1 (h) ⊗ p F 3 (h) ⊗3 + 1 12 T 5 • p E 2 (h) ⊗2 ⊗ p F 3 (h) ⊗3 + 1 5! T 6 • p E 2 (h) ⊗ p F 3 (h) ⊗5 ≥ 0. Then, considering h ′ = λp E 1 (h) + p E 2 (h) + p E 3 (h) + p F 3 (h), we have that for all λ ∈ R, λ 6 T 4 • p E 1 (h) ⊗ p F 3 (h) ⊗3 + 1 12 T 5 • p E 2 (h) ⊗2 ⊗ p F 3 (h) ⊗3 + 1 5! T 6 • p E 2 (h) ⊗ p F 3 (h) ⊗5 ≥ 0, so necessarily T 4 • p E 1 (h) ⊗ p F 3 (h) ⊗3 = 0.
Then, considering h ′ = p E 2 (h) + λp F 3 (h) for λ ∈ R, we get successively that the two other terms are null.

Likewise, we prove successively that the terms in t 11/12 are null, and then that the terms in t 23/24 are null. This yields the convergence stated in (4.4.13). Algorithm 1 may fail to yield such expansion of f for orders no lower than 10 if the hypothesis (4.4.6) is not fulfilled. Indeed for p ≥ 5, there exist p-tuples (i 1 , . . . , i p ) such that i 1 2 +• • •+ ip 2p < 1 and i 1 , . . ., i p are all even. Such tuples do not appear at orders 8 and lower, but they do appear at orders 10 and higher, for example (0, 2, 0, 0, 4) for k = 6. In such a case, we cannot use the positiveness argument to prove that the corresponding term ⊗ip is zero, and in fact, it may be not zero.

Counter

T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h)
Let us give a counter example. Consider

f : R 2 -→ R (x, y) -→ x 4 + y 10 + x 2 y 4 .
Then f ∈ A ⋆ 5 (0) and we have

E 1 = {0}, E 2 = R • (1, 0), E 3 = {0}, E 4 = {0}, F 4 = R • (0, 1). But 1 t f (t 1/4 , t 1/10 ) = 1 t t + t + t 9/10
goes to +∞ when t → 0. Now, let us give the proof of Theorem 4.4.1 for p ≥ 5. In this proof, we assume that the subspaces E 1 , . . . , E p given in Algorithm 1 satisfy the hypothesis (4.4.6).

Proof. We develop (4.4.1), which reads:

2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k k i 1 , . . . , i p t i 1 2 +•••+ ip 2p -1 T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip + o(1) =: S.
The terms such that i 1 2 + • • • + ip 2p < 1 may diverge when t → 0, so let us prove that they are in fact null. Let

α := inf i 1 2 + • • • + i p 2p : h -→ 2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =α k i 1 , . . . , i p T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ̸ ≡ 0 ,
and assume by contradiction that α < 1. Then we have for all h ∈ R d :

t 1-α S -→ t→0           2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =α k i 1 , . . . , i p T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip           ≥ 0,
by the local minimum property. Then, considering

h ′ = λ 1 p E 1 (h) + • • • + λ p p Ep (h), we have, for all h ∈ R d and λ 1 , . . . , λ d ∈ R, 2p k=2 1 k! i 1 ,...,ip∈{0,...,k} i 1 +•••+ip=k i 1 2 +•••+ ip 2p =α λ i 1 1 . . . λ ip p k i 1 , . . . , i p T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ≥ 0. (4.4.15)
Now, we fix h ∈ R d such that the polynomial in (4.4.15) in the variables λ 1 , . . . , λ p is not identically zero, and we consider k max its highest homogeneous degree, so that we have

i 1 ,...,ip∈{0,...,kmax} i 1 +•••+ip=kmax i 1 2 +•••+ ip 2p =α λ i 1 1 . . . λ ip p k max i 1 , . . . , i p T kmax • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ≥ 0.
If k max is odd, this yields a contradiction, taking

λ 1 = • • • = λ p =: λ → ±∞.
If k max is even, we consider the index l 1 such that i l 1 =: a 1 is maximal and the coefficients in the above sum with i l 1 = a 1 are not all zero. Then fixing all the λ l for l ̸ = l 1 and taking λ l 1 → ∞, we have

i 1 ,...,ip∈{0,...,kmax} i 1 +•••+ip=kmax i 1 2 +•••+ ip 2p =α i l 1 =a 1 λ i 1 1 . . . λ ip p k max i 1 , • • • , i p T kmax • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ≥ 0.
Thus, if a 1 is odd, this yields a contradiction. If a 1 is even, we carry on this process by induction : knowing l 1 , . . . , l r , we choose the index l r+1 such that l r+1 / ∈ {l 1 , . . . , l r }, the corresponding term

i 1 ,...,ip∈{0,...,kmax} i 1 +•••+ip=kmax i 1 2 +•••+ ip 2p =α i l 1 =a 1 ,...,i l r+1 =a r+1 λ i 1 1 . . . λ ip p k max i 1 , . . . , i p T kmax • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip
is not identically null and such that i l r+1 =: a r+1 is maximal. Necessarily, a r+1 is even. In the end we will find a non-zero term whose exponents i ℓ are all even which contradicts assumption (4.4.6).

Proofs of the uniform convergence and of the non-constant property

In this section we prove the additional properties claimed in Theorem 4.4.1 : the uniform convergence with respect to h on every compact set and the fact that the function g is not constant in any of its variables h 1 , . . . , h d .

Proof. First, let us prove that the convergence is uniform with respect to h on every compact set. Let ε > 0 and let R > 0. By the Taylor formula at order 2p, there exists δ > 0 such that for ||h|| < δ,

f (x ⋆ + h) -f (x ⋆ ) - 2p k=2 1 k! i 1 +•••+ip=k k i 1 , . . . , i p T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ≤ ε||h|| 2p .
Now, let us consider t → 0 and h ∈ R d with ||h|| ≤ R. Then we have:

∀t ≤ max 1, δ R 1/(2p) , ||t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)|| ≤ δ, so that 1 t [ f (x ⋆ + t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)) -f (x ⋆ ) - 2p k=2 1 k! i 1 +•••+ip=k k i 1 , . . . , i p •t i 1 2 +•••+ ip 2p -1 T k • p E 1 (h) ⊗i 1 ⊗ • • • ⊗ p Ep (h) ⊗ip ≤ ε t ||t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)|| 2p .
We proved or assumed that the terms such that i 1 2 + • • • + ip 2p < 1 are zero. We denote by g 1 (h) the sum in the last equation with the terms such that i 1 2 + • • • + ip 2p = 1 and by g 2 (h) the sum with the terms such that

i 1 2 + • • • + ip 2p > 1.
We also define a as the smallest exponent of t appearing in g 2 (h):

a := min i 1 2 + • • • + i p 2p : i 1 , . . . , i p ∈ {0, . . . , 2p}, i 1 + • • • + i p ≤ 2p, i 1 2 + • • • + i p 2p > 1 > 1.
So that:

1 t f (x ⋆ + t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)) -f (x ⋆ ) -g 1 (h) (4.4.16) ≤ t a-1 |g 2 (h)| + ε t ||t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)|| 2p .
We remark that h → g 2 (h) is a polynomial function so is bounded on every compact set. We also have:

ε t ||t 1/2 p E 1 (h) + • • • + t 1/(2p) p Ep (h)|| 2p ≤ ε(t 1/(2p) ) 2p t ||h|| 2p = ε||h|| 2p .
So (4.4.16) converges to 0 as t → 0, uniformly with respect to h on every compact set. Now let us assume that f ∈ A ⋆ p (x ⋆ ) ; we prove that the function g defined in (4.3.6) is not constant in any of its variables in the sense of (4.

3.3). Let B ∈ O d (R) adapted to the decomposition R d = E 1 ⊕ • • • ⊕ E p . We have: 1 t [f (x ⋆ + B • (t α * h)) -f (x ⋆ )] -→ t→0 g(h).
Let i ∈ {1, . . . , p} and k such that

v i := B • e i ∈ E k .
Let us assume by contradiction that g does not depend on the i th coordinate. Considering the expression of g in (4.3.6) and setting all the variables outside E k to 0, we have:

∀h ∈ E k , λ ∈ R → T 2k • (h + λv i ) ⊗2k
is constant. Then applying (4.2.1), we have:

∀h ∈ E k , T 2k • v i ⊗ h ⊗2k-1 = 0. Moreover, for h ∈ F k-1 , let us write h = h ′ + h ′′ where h ′ ∈ E k and h ′′ ∈ F k , so that T 2k • v i ⊗ h ⊗2k-1 = T 2k • v i ⊗ h ′⊗2k-1 = 0,
where we used that

∀h (3) ∈ F k-1 , T 2k • h ′′ ⊗ h (3) ⊗2k-1 = 0
following (4.4.9), and Proposition 4.8.1. Considering the definition of E k as the orthogonal complement of F k , which is defined in (4.4.9), the last equation contradicts that v i ∈ E k .

Non coercive case

The function g we obtain in Algorithm 1 is a non-negative polynomial function which is constant in none of its variables. However, this does not always guarantee that e -g ∈ L 1 (R d ), or even that g is coercive. Indeed, g can be null on an unbounded continuous polynomial curve, while the polynomial degree of the minimum x ⋆ of f is higher than the degree of g in these variables. For example, let us consider

f : R 2 → R (4.4.17) (x, y) → (x -y 2 ) 2 + x 6 .
Then f ∈ A ⋆ 3 (0) and using Algorithm 1, we get g(x, y) = (x -y 2 ) 2 , which does not satisfy e -g ∈ L 1 (R d ). In fact this case is highly degenerate, as, with

f ε (x, y) := f (x, y) + εxy 2 = x 2 + y 4 -(2 -ε)xy 2 + x 6 ,
we have that g ε (x, y) = x 2 + y 4 -(2 -ε)xy 2 satisfies e -gε ∈ L 1 (R d ) for every ε ∈ (0, 4) and that x ⋆ is not the global minimum of f ε for every ε ∈ (-∞, 0) ∪ (4, ∞).

We now prove that instead of assuming e -g ∈ L 1 (R d ), we can only assume that g is coercive, which is justified in the following proposition. More specific conditions for g to be coercive can be found in [START_REF] Bajbar | Coercive polynomials and their Newton polytopes[END_REF] and [START_REF] Bajbar | Coercive polynomials: stability, order of growth, and Newton polytopes[END_REF]. Proposition 4.4.6. Let g : R d → R be the polynomial function obtained from Algorithm 1. If g is coercive, then e -g ∈ L 1 (R d ).

Proof. Let

A k := Span (e i : i ∈ {dim(E 1 ) + • • • + dim(E k-1 ) + 1, . . . , dim(E 1 ) + • • • + dim(E k )})
for k ∈ {1, . . . , p}. By construction of g, note that for all t ∈ [0, +∞),

g p k=1 t 1/2k p A k (h) = tg(h).
Since g is coercive, there exists R ≥ 1 such that for every h with ||h|| ≥ R, g(h) ≥ 1. Then, for every h ∈ R d , we have:

g(h) = g p k=1 p A k (h) = g p k=1 ||h|| 1/2k R 1/2k p A k R 1/2k h ||h|| 1/2k = ||h|| R g p k=1 p A k R 1/2k h ||h|| 1/2k .
Then, for ||h|| ≥ R,

p k=1 p A k R 1/2k h ||h|| 1/2k 2 = p k=1 R 1/k ||h|| 1/k ||p A k (h)|| 2 ≥ R ||h|| ||h|| 2 = R||h|| ≥ R 2 ≥ R, so that g(h) ≥ ||h|| R which in turn implies e -g ∈ L 1 (R d
). We now deal with the simplest configuration where the function g is not coercive, as described in (4.4.18), by dealing with the case where f is given by (4.4.17), which is an archetype of such configuration. However, dealing with the general case is more complicated and to give a general formula for the rate of convergence of the measure π t in this case is not our current objective.

Proposition 4.4.7. Let the function f be given by (4.4.17). Then, if (X t , Y t ) ∼ C t e -f (x,y)/t dxdy, we have:

X t t 1/6 , Y 2 t -X t t 1/2 -→ t→0 C e -x 6 √ x e -y 2 √ π 1 x≥0 dxdy, where C = ∞ 0 e -x 6 √ x dx -1 .
Proof. First, let us consider the normalizing constant C t . We have :

C -1 t = R 2 e -(x-y 2 ) 2 +x 6 t dxdy = 2t 3/4 ∞ -∞ e -t 2 x 6 ∞ 0 e -(y 2 -x) 2 dy dx = t 3/4 ∞ -∞ e -t 2 x 6 ∞ -x e -u 2 √ u + x dy dx = t 7/12 ∞ -∞ e -x 6 ∞ -t -1/3 x e -u 2 t 1/3 u + x du dx ∼ t→0 t 7/12 ∞ 0 e -x 6 √ x ∞ -∞ e -u 2 du dx,
where the convergence is obtained by dominated convergence and where we performed the change of variables x ′ = t -1/6 x and u = t -1/2 (y 2 -x). Then we consider, for a 1 < b 1 and a 2 < b 2 ,

P X t t 1/6 ∈ [a 1 , b 1 ], Y 2 -X t 1/2 ∈ [a 2 , b 2 ] .
Performing the same changes of variables and using the above equivalent of C t completes the proof.

More generally, if the function g is not coercive and if we can write, up to a change of basis,

g(h 1 , . . . , h d ) = Q 1 (h 1 , h 2 ) 2 + Q 2 (h 3 , h 4 ) 2 + • • • + Q r (h 2r-1 , h 2r ) 2 + g(h 2r+1 , . . . , h d ), (4.4.18)
where the Q i are polynomials with two variables null on an unbounded curve (for example,

Q i (x, y) = (x -y 2 ), Q i (x, y) = (x 2 -y 3 ), Q i (x, y) = x 2 y 2 )
, and where g is a non-negative coercive polynomial, then a 1 ((X t ) 1 , (X t ) 2 , t) , . . . , a r ((X t ) 2r-1 , (X t ) 2r , t) ,

1 t α 2r+1 , . . . , 1 t α d * B • ((X t ) 2r+1 , . . . , (X t ) d ) -→ t→0 b 1 (x 1 , x 2 ) . . . b r (x 2r-1 , x 2r )Ce -g(x 2r+1 ,...,x d ) dx 1 . . . dx 2r dx 2r+1 • • • dx d ,
where C is a normalization constant, B ∈ O d-2r-1 (R) is an orthogonal transformation and for all k = 1, . . . , r, a k : R 2 × (0, +∞) → R 2 and b k is a density on R 2 . Such a k and b k can be obtained by applying the same method as in Proposition 4.4.7. Algorithm 1 yields the first change of variable for this method, given by the exponents (α i ) (in the proof of Proposition 4.4.7, the first change of variable is t -1/2 x and t -1/4 y) and thus seems to be the first step of a more general procedure in this case. However, we do not give a general formula as the general case is cumbersome. Moreover, we do not give a method where the non coercive polynomials Q i depend on more than two variables, like

Q(x, y, z) = (x -y 2 ) 2 + (x -z 2 ) 2 .
The method sketched in Proposition 4.4.7 cannot be direclty applied to this case.

Proofs of

Single well case

We now prove Theorem 4.3.3.

Proof. Using Theorem 4.4.1, we have for all h ∈ R d :

1 t [f (x ⋆ + B • (t α * h)) -f (x ⋆ )] -→ t→0 g(h).
To simplify the notations, assume that there is no need of a change of basis i.e. B = I d . We want to apply Theorem 4.3.2 to the function f . However the condition

R d sup 0<t<1 e -f (x ⋆ +(t α 1 h 1 ,...,t α d h d ))-f (x ⋆ ) t dh 1 . . . dh d < ∞
is not necessarily true. Instead, let ε > 0 and we apply Theorem 4.3.2 to f , where f is defined as:

f (h) = f (h) if h ∈ B(x ⋆ , δ) ||h -x ⋆ || 2 + f (x ⋆ ) else,
and where δ > 0 will be fixed later. Then f satisfies the hypotheses of Theorem 4.3.2. The only difficult point to prove is the last condition of Theorem 4.3.2. If t ∈ (0, 1] and h ∈ R d are such that (t

α 1 h 1 , . . . , t α d h d ) / ∈ B(0, δ), then f (x ⋆ + (t α 1 h 1 , . . . , t α d h d )) -f (x ⋆ ) t = ||(t α 1 h 1 , . . . , t α d h d )|| 2 t ≥
, . . . , t α d h d ) ∈ B(0, δ), f (x ⋆ + (t α 1 h 1 , . . . , t α d h d )) -f (x ⋆ ) t -g(h) ≤ ε,
which is possible because of the uniform convergence on every compact set (see Section 4.4.9), we derive that

f (x ⋆ + (t α 1 h 1 , . . . , t α d h d )) -f (x ⋆ ) t ≥ g(h) -ε. Hence R d sup 0<t<1 e -f (x ⋆ +t α 1 h 1 ,...,t α d h d ))-f (x ⋆ ) t dh 1 . . . dh d ≤ R d e -||h|| 2 dh + e ε R d e -g(h) dh.
Since g is coercive, using Proposition 4.4.6 we have e -g ∈ L 1 (R d ) and it follows from Theorem 4.3.2 that if X t has density π t (x) := C t e -f (x)/t , then

( X t -x ⋆ ) 1 t α 1 , . . . , ( X t -x ⋆ ) d t α d L -→ X as t → 0,
where X has density proportional to e -g(x) . Now, let us prove that if X t has density proportional to e -f (x)/t , then we also have

(X t -x ⋆ ) 1 t α 1 , . . . , (X t -x ⋆ ) d t α d L -→ X as t → 0. (4.5.1) Let φ : R d → R be continuous with compact support. Then E φ (X t ) 1 t α 1 , . . . , (X t ) d t α d -φ ( X t ) 1 t α 1 , . . . , ( X t ) d t α d = R d φ x 1 t α 1 , . . . , x d t α d C t e -f (x 1 ,...,x d ) t -C t e -f (x 1 ,...,x d ) t dx 1 . . . dx d =: I 1 + I 2 ,
where I 1 is the integral on the set B(x ⋆ , δ) and I 2 on B(x ⋆ , δ) c . We have then:

|I 2 | ≤ ||φ|| ∞ (π t (B(x ⋆ , δ) c ) + π t (B(x ⋆ , δ) c )) -→ t→0 0,
where we used Proposition 4.3.1. On the other hand, we have f = f on B(x ⋆ , δ), so that

|I 1 | ≤ ||φ|| ∞ |C t -C t | B(x ⋆ ,δ) e -f (x) t dx ≤ ||φ|| ∞ 1 - C t C t .
And we have:

C t C t = e -f (x) t dx e -f (x) t dx = B(x ⋆ ,δ) e -f (x) t dx + B(x ⋆ ,δ) c e -f (x) t dx B(x ⋆ ,δ) e -f (x) t dx + B(x ⋆ ,δ) c e -f (x) t dx
.

By Proposition 4.3.1, we have when t → 0

B(x ⋆ ,δ) c e -f (x) t dx = o B(x ⋆ ,δ) e -f (x) t dx B(x ⋆ ,δ) c e -f (x) t dx = o B(x ⋆ ,δ) e -f (x) t dx ,
so that C t /C t → 1, so I 1 → 0, which then implies (4.5.1).

Multiple well case

We now prove Theorem 4.3.5.

Proof. The first point is a direct application of Theorem 4.3.4. For the second point, we remark that X it has a density proportional to e -f i (x)/t , where

f i (x) := f (x) if x ∈ B(x ⋆ i , δ) +∞ else.
We then consider f i as in Section 4.5.1:

f i (x) = f i (x) if x ∈ B(x ⋆ i , δ) ||x -x ⋆ i || 2 + f ⋆ else.
and still as in Section 4.5.1, we apply Theorem 4.3.2 to f i and then prove that random variables with densities proportional to e -f i (x)/t and e -f i (x)/t respectively have the same limit in law.

Infinitely flat minimum

In this section, we deal with an example of infinitely flat global minimum, where we cannot use a Taylor expansion.

Proposition 4.6.1.

Let f : R d → R such that ∀x ∈ B(0, 1), f (x) = e -1
||x|| 2 and ∀x / ∈ B(0, 1), f (x) > a for some a > 0. Furthermore, assume that f is coercive and e -f ∈ L 1 (R d ). Then, if X t has density π t ,

log 1/2 1 t • X t L -→ X as t → 0,
where X ∼ U(B(0, 1)).

Proof. Noting that ||x||>1 e -f (x)/t dx → 0 as t → 0 by dominated convergence, we have

C t ∼ t→0 B(0,1) e -e -1 ||x|| 2 /t dx -1 = log d/2 1 t         B(0, √ log(1/t)) e -t 1 ||x|| 2 -1 dx → t→0 Vol(B(0,1))         -1
, where the convergence of the integral is obtained by dominated convergence. Then we have, for -1 < a i < b i < 1 and i a 2 i < 1, i b 2 i < 1:

P log 1/2 1 t • X t ∈ d i=1 [a i , b i ] = C t log d/2 1 t (b i ) (a i ) e -t 1 |x| 2 -1 dx -→ t→0 d i=1 (b i -a i ) Vol(B(0, 1)) .

Simulations: computing high-order expansion of the loss with singular Hessian matrix

In this section, we present simulations illustrating our theoretical results for degenerate cases and showing how we can compute such high-order expansions in practice. Our code and a demonstration notebook are available at https://github.com/Bras-P/gibbs-measures-with-singular-hessian.

We consider the function f to be the loss function associated to the training of an artificial neural network. We train a feedforward neural network on the MNIST dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], which is composed of grayscale images of size 28 × 28 of handwritten digits (from 0 to 9). The neural network is composed of two hidden layers with 16 units each and with ReLU activation. After training the neural network on the data, we compute the Hessian matrix of the loss with respect to the kernel of the second Dense layer, as in practice we cannot compute the whole Hessian matrix with respect to all the variables. The Hessian matrix is singular; we give the distribution of its eigenvalues in Figure 4.1 where a gap appears between non-zero eigenvalues and zero eigenvalues.

We then compute the expansion of the loss function as in (4.4.2) up to the order 4 on the subspace of dimension 2 which is spanned by two eigenvectors, one with non-zero eigenvalue (subspace E) and one with zero eigenvalue (subspace F ).

In TensorFlow, higher order derivatives can be computed the same way as first-order derivatives by stacking several tf.GradientTape. We use tf.GradientTape.jacobian to compute the Jacobian tensor of a tensor-valued function, since tf.GradientTape.gradient only computes the gradient of a scalar-valued function. Since the dimension of the variable may be very large for neural network, we compute derivative tensors only with respect to some variables. To do so, we need to create a new Layer object where the reduced kernel only includes these variables. We give the implementation of a dense layer modified in order to compute (t 1 , . . . , t r ) ∈ R r → ℓ(x ⋆ + t 1 v 1 + • • • + t r v r ) where ℓ stands for the output of the layer, where x ⋆ ∈ R d is fixed and where v 1 , . . . , v r are vectors. The new layer class is defined as subclass of the base class tf.keras.layers.Layer. The method build initializes the weights of the layer when the layer is called for the first time; each weight is defined through the method add_weight. The method call defines the output of the layer depending on the inputs and its weights. We then obtain the following expansion up to the order 4; if we denote by x ⋆ the (empirical) minimum of the loss function, by x 1 the eigenvector with non-zero eigenvalue and by x 2 the eigenvector with zero eigenvalue, for λ 1 , λ 2 ∈ R we have

1 t f (x ⋆ + t 1/2 λ 1 x 1 + t 1/4 λ 2 x 2 ) -f (x ⋆ ) -→ t→0 λ 2 1 2 ∇ 2 f (x ⋆ ) • x ⊗2 1 + λ 1 λ 2 2 2 ∇ 3 f (x ⋆ ) • x 1 ⊗ x ⊗2 2 + λ 4 2 4! ∇ 4 f (x ⋆ ) • x ⊗4 2
where the three coefficients

∇ 2 f (x ⋆ ) • x ⊗2 1 , ∇ 3 f (x ⋆ ) • x 1 ⊗ x ⊗2 2 and ∇ 4 f (x ⋆ ) • x ⊗4
2 are computed in the notebook by stacking several tf.GradientTape as in compute_hessian.

( < tf . Tensor : shape =() , dtype = float32 , numpy =0.026905548 > , < tf . Tensor : shape =() , dtype = float32 , numpy = -0.0007531713 > , < tf . Tensor : shape =() , dtype = float32 , numpy = -3.2213422 e -05 >) 4.8. Appendix: Properties of tensors Proof. Using (4.2.1), we have for h 1 , . . . , h k ∈ E and λ 1 , . . . , λ k ∈ R,

T k • (λ 1 h 1 + • • • + λ k h k ) ⊗k = i 1 +•••+i k =k k i 1 , . . . , i k λ i 1 1 . . . λ i k k T k • h ⊗i 1 1 ⊗ • • • ⊗ h ⊗i k k = 0,
which is an identically null polynomial in the variables λ 1 , . . . , λ k , so every coefficient is null, in particular ∀h 1 , . . . ,

h k ∈ E, T k • h 1 ⊗ • • • ⊗ h k = 0.

Part II

Adaptive Langevin algorithms for deep Neural Networks

Chapter 5

Stochastic algorithms for artificial neural networks with simulations in TensorFlow

Artificial neural networks

Calibration of artificial neural networks as a stochastic optimization problem

Neural networks are a powerful tool for a wide range of problems (regression, classification, etc) involving inferring a model from a huge amount of data. Neural networks are functions with high-dimensional parametrization thus allowing to fit to many different problems and data. With neural networks, the user does not need to devise a very specific model fitting to the data by himself anymore. Let us consider data samples u i ∈ R d in and y i ∈ R dout for 1 ≤ i ≤ M associated to a regression problem, where (u i ) are the inputs and where (y i ) are the outputs. That is, we look for a function ψ : R d in → R dout which fits to the data i.e.

∀ 1 ≤ i ≤ M, ψ(u i ) ≈ y i .
In other words, the objective is to extract a model from the empirical data. We look for a function ψ in a family of functions parametrized by a finite-dimensional parameter: {ψ x , x ∈ R d }. For L : R dout → R a loss function which measures the error between the prediction ψ x (u i ) and the true data y i , the regression problem then becomes the minimization of the average loss over the data and can be written as the following optimization problem:

Minimize x∈R d V (x) := 1 M M i=1 L(ψ x (u i ) -y i ). (5.1.1)
The output of a fully connected neural network ψ x is defined as follows. Let us choose an activation function φ : R → R, which is a non-linear sigmoid-type or ReLU-type function (see Table 5.1). Let K + 1, K ∈ N, be the number of layers of the neural network and for k = 0, . . . , K let d k ∈ N be the size of the k th layer with d 0 = d in and d K = d out ; the output is defined recursively as: The second equation should be understood for each coordinate of u (k) . For every 1

u (0) := u ∈ R d in , u (k) = φ(α k • u (k-1) + β k ) ∈ R d k , 1 ≤ k ≤ K -1, ψ x (u) = u (K) = α K • u (K-1) + β K .
≤ k ≤ K, α k ∈ M d k ,d k-1 (R) and β k ∈ R d k and the parameter x of the neural network is x = (α 1 , β 1 , . . . , α K , β K ) ∈ M d 1 ,d 0 (R) × R d 1 × • • • × M d K ,d K-1 (R) × R d K .
Neural networks have first gained interest as universal approximators since Cybenko's theorem [START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF] stating that if

d out = 1, if K = 2 and if φ : R → R is continuous and such that φ(t) -→ t→-∞ 0 and φ(t) -→ t→+∞ 1 then the set    u → d 1 j=1 α j 2 φ([α 1 • u + β 1 ] j ), d 1 ∈ N, α 1 ∈ M d 0 ,d 1 (R), β 1 ∈ R d 1 , α 2 ∈ R d 1    is dense in C 0 ([0, 1] d in , R)
for the uniform norm. In fact, no monotonicity assumption is required for φ.

Neural networks architectures for tasks other than regression

Classification

Let us consider a classification problem: the output to predict y is not a "continuous" data having values in R dout but is a classification label in {1, . . . , N labels }. In this case, the output y is mapped to R N labels as a one-hot vector i.e. for 1 ≤ j ≤ N labels , the j th coordinate is the probability to belong to the j th class. Similarly to the logistic classification, the loss function used for classification problems is the categorical cross-entropy: if the data y = i ∈ {1, . . . , N labels } and the prediction vector is z ∈ R N labels , then the resulting loss is log(1 + e -z i ). For more details we refer to [START_REF] Tsoumakas | Mining Multi-label Data[END_REF].

Images and convolutional layers

An image encoded as a matrix can be seen as an input vector, however this does not take into account the spatial properties of the image. Moreover, the original dimension of an image is generally too large to be directly processed using fully-connected layers. A common way to deal with image analysis is to stack a succession of 2D-convolutions which kernel is a trainable parameter in order to extract its spatial features, and of pooling layers which are non-trainable but reduce the dimension of the image by averaging or taking the maximum over small squares 5.1. Artificial neural networks

RGB input image multi-channel image

Conv2D 2×2 pool

Figure 5.2: 2D-convolution layer followed by max-pooling architecture [START_REF] Jarrett | What is the best multi-stage architecture for object recognition?[END_REF]. Once the features are extracted and the dimension is reduced, we use dense layers to extract the role of each feature. An illustration is given in Figure 5.2.

Time series

The problem is to predict the value of a time series in the future knowing the previous values over a time window. This problem can be rewritten as a regression problem as follows. For a whole time series (u k ) 1≤k≤T ∈ (R d 1 ) T , the input data is the extracted sub-series (u k+1 , . . . , u k+ℓ ) where ℓ ≪ T and for each k the output is u k+ℓ+s where s is the number of time steps in the future for which we predict the value of u.

To design a neural network adapted to time series analysis, we first need layers that extract the features of the data and that take into account their sequential structure. Along with images, we can use 1D-convolutional layers however such layers only extract short-term local features. Recurrent Neural Networks (RNN) are more adapted to the analysis of short but also long-term dependencies. RNN have a "hidden state" which is a time series that is recursively updated according to the data [START_REF] Rumelhart | Learning representations by backpropagating errors[END_REF]. More specifically, the output of a RNN is the hidden state (h k+1 , . . . , h k+ℓ ) that is recursively updated as

h j+1 = ψ x (u j , h j )
where ψ x is some neural function with trainable parameter x (independent of j). Popular choices for the parametrization of ψ x are LSTM (Long Short Term Memory) networks [START_REF] Hochreiter | LSTM can solve hard long time lag problems[END_REF][START_REF] Hochreiter | Long short-term memory[END_REF] and GRU (Gated Recurrent Unit) networks [START_REF] Cho | On the properties of neural machine translation: Encoder-decoder approaches[END_REF].

Neural networks, stochastic gradient and automatic differentiation

The stochastic gradient descent for (5.1.1) writes

x n+1 = x n -γ n+1 ∇ x L(ψ xn (u i n+1 ) -y i n+1 ) (5.1.2)
where i n+1 ∈ {1, . . . , M } is an index chosen uniformly at random and where (γ n ) is a nonincreasing step sequence. In practice, SGD is implemented by batches where the gradient is estimated by the average over a small amount of data:

x n+1 = x n - γ n+1 N batch i∈I n+1 ∇ x (L(ψ xn (u i ) -y i )) (5.1.3)
where I n+1 is a subset of {1, . . . , M } of size N batch taken uniformly at random with N batch ≪ M . The advantages of using batches are that the gradient descent is more stable and that for each iteration, all the gradients ∇ x (L(ψ xn (u i ) -y i )) for i ∈ I n+1 are computed simultaneously by parallelization in particular using GPU devices, so that computing N batch gradients takes the same time as computing one gradient provided that N batch is not too large.

Computing the gradient ∇ x (L(ψ xn (u i ) -y i )) using the finite difference method is inaccurate and cumbersome in particular when the dimension of the parameter x is large. Instead we use automatic differentiation: knowing that L(ψ x (u i ) -y i ) is written as a composition of elementary operations (addition, multiplication, etc) and functions (exp, log etc) of the parameter x as it is the case for neural networks, then the gradient with respect to x is explicitly computed using the chain rule.

In a more general framework, let us consider a function f : R d → R q of x which can be written as a composition of elementary operations which gradients are analytically known. More specifically, let us write these elementary operations as f (x 0 ) = x N with x N defined recursively as

x 0 ∈ R ℓ 0 , x n+1 = x n f n+1 (x n ) =: F n+1 (x n ) ∈ R ℓ n+1 , n ∈ {0, . . . , N -2}, x N = f N (x N -1 ) (5.1.4)
where f n+1 : R ℓn → R ℓ n+1 -ℓn and F n+1 : R ℓn → R ℓ n+1 and d = ℓ 0 < ℓ 1 < • • • < ℓ N -1 are increasing dimensions and ℓ N = q. Thus in order to compute the gradient ∂x N /∂x 0 , writing ẋn := ∂x n /∂x 0 ∈ R ℓn×ℓ 0 with ẋ0 = I ℓ 0 the identity matrix and using the chain rule we have ẋn+1 = ∇F n+1 (x n ) • ẋn .

(5.1.5) However (5.1.5) uses the matrix products ∇F n+1 (x n )• ẋn involving matrix products of dimensions (ℓ n+1 × ℓ n ) • (ℓ n × ℓ 0 ), which is computationally expensive in the cases where ℓ 0 (the number of parameters with respect to which we compute the gradient) is large and ℓ N is small (typically ℓ N = 1). Another method is to consider instead the adjoint

xn := ∂x N ∂x n ⊤ ∈ R ℓn×ℓ N (5.1.6)
with xN = I ℓ N so that we have the backward recursive relation

xn = (∇F n+1 (x n )) ⊤ • xn+1 , (5.1.7) involving matrix products of dimensions (ℓ n × ℓ n+1 ) • (ℓ n+1 × ℓ N ).
Thus a method to compute ẋN = x0 is:

1. Forward step: compute x N and the gradients ∇F n+1 (x n ) using the forward recursive relation (5.1.5).

2. Backward step: compute x0 using the backward recursive relation (5.1.7).

The gradients ∇F n+1 (x n ) are directly computed using the known analytical expression of ∇f n+1 for f n+1 in a finite set of elementary operations and functions. For example, for f (x) = e x then ∇f (x) = e x ; for f (x, y) = x + y ∈ R ℓ then ∇f (x, y) = (I ℓ , I ℓ ); for f (x, y) = x * y ∈ R ℓ then ∇f (x, y) = (diag(y), diag(x)).

For more details we refer to [START_REF] Michael | Monte Carlo evaluation of sensitivities in computational finance[END_REF][START_REF] Güneş Baydin | Automatic differentiation in machine learning: a survey[END_REF].

x 0 x ⋆ V Figure 5
.3: Behaviour of SGD when the learning rate is too large.

Stochastic gradient algorithms

Adaptive stochastic algorithms

Many variants of the the classic SGD algorithm (5.1.2) have been developed in order to make more efficient and more stable algorithms and to accelerate their speed of convergence. One main drawback of (5.1.2) is that the learning rate (γ n ) must be set by the user and may not be adapted simultaneously to any optimization problem (5.1.1) in general. Indeed, if the learning rate is too large then the algorithm may be repelled from minima or diverge as illustrated in Figure 5.3. Other stochastic gradient methods are usually based on adaptive learning rates and for a noisy measurement of the gradient g n+1 they read

x n+1 = x n -γ n+1 P n+1 • g n+1 (5.2.1)
where P n+1 ∈ M d (R) is positive definite. As the dimension of x is large, P n+1 is chosen to be a diagonal matrix, thus yielding an adaptive learning rate for each dimension. Each different choice for the preconditionner rule P yields a different algorithm; let us present some of the most used stochastic gradient methods in practice.

The RMSprop algorithm [START_REF] Tieleman | Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude[END_REF] consists in maintaining a discounted average of the square of gradients for each weight and to divide the gradient by the root of this average. More precisely, given α ∈ (0, 1) close to 1 and λ > 0 close to 0 the RMSprop update is written in Algorithm 1.

Algorithm 1 RMSprop for 1 ≤ n ≤ N iter do g n+1 = ∇ x L(ψ xn (u i n+1 ) -y i n+1 ) MS n+1 = α MS n +(1 -α)g n+1 ⊙ g n+1 P n+1 = diag 1 ⊘ λ1 + √ MS n+1 x n+1 = x n -γ n+1 P n+1 • g n+1 end for
The parameter λ is used for numerical stability. The operators ⊙ and ⊘ represent elementwise product and division, respectively. Using a discounted average with rate α instead of the current value allows to counter the unstable mini-batch effects, as evaluating the gradient on mini-batches leads to estimates that may greatly differs from one iteration to the next. The Adam algorithm [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] is based on first and second moment estimates of the gradient. For β 1 ∈ (0, 1) and β 2 ∈ (0, 1) both close to 1 and for λ > 0 close to 0, the Adam update is written in Algorithm 2. Note that in the update of x n+1 , an average of the gradient M n+1 over the past iterations with exponentially decreasing weight is considered. The notations β n+1 1 and β n+1 2 denote β 1 and β 2 to the power n + 1 respectively.

Algorithm 2 Adam for 1 ≤ n ≤ N iter do g n+1 = ∇ x L(ψ xn (u i n+1 ) -y i n+1 ) M n+1 = β 1 M n + (1 -β 1 )g n+1 MS n+1 = β 2 MS n +(1 -β 2 )g n+1 ⊙ g n+1 M n+1 = M n+1 /(1 -β n+1 1 ) MS n+1 = MS n+1 /(1 -β n+1 2 ) P n+1 = diag 1 ⊘ λ1 + MS n+1 x n+1 = x n -γ n+1 P n+1 • M n+1 . end for
The Adadelta algorithm [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] uses as preconditioner the moving root mean square of the increments of (x n ) divided by the moving root mean square of the gradients (g n ) and is given in Algorithm 3.

Algorithm 3 Adadelta for 1 ≤ n ≤ N iter do g n+1 = ∇ x L(ψ xn (u i n+1 ) -y i n+1 ) MS n+1 = β 1 MS n +(1 -β 1 )g n+1 ⊙ g n+1 P n+1 = diag (λ1 + M S n ) ⊘ λ1 + MS n x n+1 = x n -γ n+1 P n+1 • g n+1 . M S n+1 = β 2 MS n +(1 -β 2 )(x n+1 -x n ) ⊙ (x n+1 -x n ).

end for

Other optimizers used in Machine Learning mainly rely on the same idea which is to adapt the learning rate for each weight using gradient moving averages and are often based on tunes of one of the previous algorithms, such as Adamax [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF], which is a variant of Adam based on the infinity norm instead of the L 2 -norm, and AMSgrad [START_REF] Sashank | On the convergence of adam and beyond[END_REF].

Likewise, vanilla or preconditioned Langevin algorithms are obtained by adding Gaussian noise to a stochastic gradient method for some choice of preconditioner. More specifically, the Langevin version of (5.2.1) reads

x n+1 = x n -γ n+1 P n+1 • g n+1 + a n+1 γ 1/2 n+1 P 1/2 n+1 • N (0, I d ) + a 2 n+1 γ n+1 Υ n+1 , (5.2.2)
where a n+1 is a scalar positive decreasing or constant sequence controlling the amplitude of the noise and where Υ is a second-order correction term given by

Υ n+1 1≤i≤d = d j=1 ∂ j [P n+1 ] ij 1≤i≤d . ( 5

Gradient optimizers in TensorFlow

We Then the model is trained and its weights are fitted to the data: model . fit ( train_features , train_labels , batch_size =256 , epochs =20 , validation_data =( val_features , val_labels ) )

Some gradient algorithms are already implemented in TensorFlow and can be found in the module tf.keras.optimizers, such as tf.keras.optimizers.SGD, tf.keras.optimizers. Adam, tf.keras. optimizers.RMSprop. We implement Langevin algorithms as subclasses of the base class tf.keras. optimizers.Optimizer. The method _resource_apply_dense(self, grad, var) defines the algorithm update rule applied to the weight of the model var knowing the derivative of the loss grad with respect to var. For example, for the basic stochastic gradient algorithm, the update rule is simply Auxiliary variables of the optimizers can be defined in create_slots(self, var_list), for example the gradient square moving average MS n for RMSprop, see Algorithm 1.

We give the code using these methods for the L-RMSprop algorithm in Listing 5.2. The code for other L-algorithms can be found at https://github.com/Bras-P/langevin-simulated-annealing/blob/main/optimizers.py.

We first create a subclass LangevinOptimizer of the base TensorFlow optimizer class tf.keras. optimizers.Optimizer in order to handle the possible decreasing of sigma which is either a float value either a tf.keras.optimizers. schedules. LearningRateSchedule: The correction term Υ which includes second-order derivatives can be computed using automatic differentiation as same as for the computation of first-order derivatives with the function diag_jacobian from the module tensorflow_probability.python.math. However in general we do not include the correction term Υ in the Langevin optimizers as computing second-order derivatives highly increases the computation time and including the term Υ does not lead to any visible improvement in practice in comparison with the computation time. Moreover the time step coefficient of Υ n in (5.2.2) is a 2 n γ n with a n ≪ 1 and γ n ≪ 1, in comparison with γ n for the gradient and a n γ 1/2 n for the Gaussian noise, and if the regularization coefficient α is close to 1 then the preconditioner slowly varies at each iteration and then the true correction term Υ is close to 0.

Simulations

We compare the performances of various adaptive (preconditioned) Langevin algorithms with vanilla SGLD, which is the Langevin algorithm with constant (additive) σ (P n ), as well as with standard non-Langevin algorithms. The algorithms are tested on diverse optimization and inference problems with real-life data.

Double well potential

Let us start with a simple one-dimensional example. Let V : R → R be a coercive C 2 "doublewell"-type function i.e. with two local minima x ⋆ 1 and x ⋆ 2 and such that V (x ⋆ 1 ) ≥ V (x ⋆ 2 ). In this section we consider

V (x) = 1 4 x 4 - 1 2 x 2 -cx,
see Figure 5.4. Starting in the the neighbourhood of x ⋆ 1 = -1, we simulate how many iterations are needed to escape the first local minimum and to reach x ⋆ 2 = 1. The Gaussian noise added to the gradient descent indeed allows to escape from local minima and to explore the whole space. The results are given in Table 5.1, where we compare the performances of preconditioned Langevin algorithms with vanilla SGLD. Preconditioned Langevin algorithms show better performances in exploring the state space with a lower mean escape time.

Bayesian Logistic Regression

We compare vanilla SGLD with preconditioned Langevin algorithms on a Bayesian logistic regression problem. Given a binary classification problem (u i , y i ) 1≤i≤N where y i ∈ {±1}, the probability of the i th output is

p(y i |u i , x) = 1 1 + exp (-y i ⟨x, u i ⟩) ,
corresponding to the binary entropy loss. We use the Adult dataset a9a [START_REF] Kohavi | Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid[END_REF] where the binary classification task is to determine whether a person earns over 50k$ per year knowing 14 features such as age, education etc. We use a Bayesian approach for this problem and we simulate the parameter x according to the Gibbs distribution of density proportional to e -2V (x)/a 2 . The empirical posterior distribution is given by the N last values of the weights in the algorithm. The advantage of the Bayesian approach is to be able to quantify the uncertainty about the inferred values of the weights and about the underlying process.

The results are given in Figure 5.5, showing that the decrease of the loss function is faster for precondtioned Langevin algorithms than for vanilla SGLD.

Image classification (1)

We train a neural network on the MNIST dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF], which is composed of grayscale images of size 28×28 of handwritten digits (from 0 to 9). The goal is to recognize the handwritten digit and to classify the images. 60.000 images are used for training and 10.000 images are used for test. Examples of predictions are given in Figure 5.6.

We first compare the performances of various Langevin algorithms for the training of a feedforward neural network, composed of two hidden dense layers with 128 units each and with ReLU activation. The results are given in Figure 5.7. The preconditioned Langevin algorithms show significant improvement compared with the vanilla SGLD algorithm. Their convergence is faster and they achieve a lower error on the test set. We also display the value of the loss function on the train set during the training to show that the better performances of the preconditioned algorithms are not due to some overfitting effect.

We then compare preconditioned Langevin algorithms with their respective non-Langevin counterparts. For shallow neural networks, adding an exogenous noise does not seem to improve We display the performance of SGD for reference. The values of the hyperparameters are a(n) = A log -1/2 (c 1 n+ e) with A = 1.10 -3 for L-Adam and A = 5.10 -4 for L-RMSprop and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n) where c 2 n = 1 after 5 epochs and where γ 1 = 0.01 for SGLD and γ 1 = 0.001 for the others. the performances of the optimization algorithm, although it adds some regularization. In particular, we could not reproduce the good results of [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] for feedforward neural networks, as it is also noted in a footnote in [START_REF] Marceau-Caron | Natural Langevin Dynamics for Neural Networks[END_REF]. However, for deep neural networks, which are highly non-linear and which loss function has many local minima, the Langevin version is competitive with the currently widely used non-Langevin algorithms and can even lead to improvements. The results are given in Figure 5.8 where we used a deep neural network with 20 hidden layers of 32 units each and with ReLU activation. This aspect shall be developed in a Chapter 6.

Image classification (2)

We train a convolutional neural network on the CIFAR-10 dataset [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF], which is composed of RGB images of size 32 × 32 belonging to 10 different classes: airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks. 50000 images are used for training and 10000 images for test. We train a deep neural network composed of two convolutional layers with 4 × 4 kernel size and 32 channels for each; 2 × 2 max-pooling is used after each convolutional layer, which is a standard network configuration [START_REF] Jarrett | What is the best multi-stage architecture for object recognition?[END_REF]. These layers are followed by 20 hidden dense layers with 64 units each and with ReLU activation. Since the images in the CIFAR10 dataset do not have a good resolution, we cannot expect a very high accuracy on the test set.

We proceed to side-by-side comparison of Langevin algorithms with their respective non-Langevin counterparts. The results are given in Figure 5.9 and show that in the case of a deep neural network with a large number of hidden layers, preconditioned Langevin optimizers achieve competing or even faster convergence speed than non-Langevin optimizers. For more developments we refer to Chapter 6.

Time series analysis and prediction

We use the Jena Climate dataset which is a weather time series dataset recorded at the weather station of the Max Planck Institute for Biogeochemistry in Jena, Germany. This dataset contains 14 different features such as air temperature, atmospheric pressure, humidity and wind. We also pre-process the dataset and add some relevant features for weather prediction such as the the We compare the performance of the RNN model with the performance of the baseline model which is the model obtained by simply predicting the future value by the current value. An example of prediction of the temperature using a RNN is given in Figure 5.10.

We compare the performances of the preconditioned Langevin algorithms with the vanilla SGLD algorithm in Figure 5.11 and in Table 5

(n) = A log -1/2 (c 1 n + e)
with A = 2.10 -3 and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n) where c 2 n = 1 after 5 epochs and where for SGLD, γ 1 = 0.001 for L-RMSprop and L-Adam and γ 1 = 0.1 for L-Adadelta.

Optimal Quantization

We consider the problem of vector optimal quantization as stated in Section 1.1.2.5, which does not involve neural networks. For µ some probability distribution in L 2 (R q ) and for fixed K ∈ N, the objective is to minimize the distortion function:

min x=(x 1 ,...,x K )∈(R q ) K V (x) := 1 2 R q min 1≤k≤K |ξ -x k | 2 µ(dξ).
(5.3.1)

Then defining the Voronoï partition of R q :

V k (x 1 , . . . , x K ) := ξ ∈ R q ∀ 1 ≤ j ≤ K, |ξ -x k | ≤ |ξ -x j | , 1 ≤ k ≤ K,
the corresponding gradient descent algorithm reads with X n = (X 1 n , . . . , X K n ) ∈ (R q ) K and Y n ∼ µ and iid:

X n+1 = X n -γ n+1 1 Y n+1 ∈V k (Xn) (X k n -Y n+1 ) 1≤k≤K . (5.3.2)
Optimal quantization problems involve the minimization of a potential function (distortion) which is highly non-convex and with many local minima and saddle points. Indeed, if x = (x 1 , . . . , x K ) ∈ (R q ) K is a global minimizer, then by symmetry and for every τ ∈ S K the group of permutations of {1, . . . , K}, (x τ (1) , . . . , x τ (K) ) is also a minimizer. It follows from the Mountain Pass theorem [START_REF] Ambrosetti | Dual variational methods in critical point theory and applications[END_REF] the existence of critical points which are not global minimizer. Moreover, the set of global minimizers is even larger if the law µ to be quantified has its own symmetry properties. Typically for µ = N (0, I q ), if x = (x 1 , . . . , x K ) ∈ (R q ) K is a global minimizer then for every linear isometry B, (Bx 1 , . . . , Bx K ) is also a global minimizer. We also refer to [GL00, Section 5] for a discussion on the uniqueness of optimal quantizers (up to permutation) in the one-dimensional case. An illustration in a simple case is given in Figure 5.13. Therefore, Langevin algorithms may be suitable for improving the CLVQ algorithm. In Figure 5.12, we consider the optimal quantization problem with µ = N (0, I q ) in dimension q = 20 with K = 1 e 4 quantization points and we proceed to side-by-side comparison of Langevin and non-Langevin optimizers, showing that Langevin methods improve the performances of classic gradient descent methods for optimal quantization. We plot the two-dimensional projection of the quantizers in Figure 5.14. Chapter 6

Langevin algorithms for very deep Neural Networks with application to image classification

The

Abstract

Training a very deep neural network is a challenging task, as the deeper a neural network is, the more non-linear it is. We compare the performances of various preconditioned Langevin algorithms with their non-Langevin counterparts for the training of neural networks of increasing depth. For shallow neural networks, Langevin algorithms do not lead to any improvement, however the deeper the network is and the greater are the gains provided by Langevin algorithms. Adding noise to the gradient descent allows to escape from local traps, which are more frequent for very deep neural networks. Following this heuristic we introduce a new Langevin algorithm called Layer Langevin, which consists in adding Langevin noise only to the weights associated to the deepest layers. We then prove the benefits of Langevin and Layer Langevin algorithms for the training of popular deep residual architectures for image classification. [START_REF] Hochreiter | Investigations on dynamic Neural Networks[END_REF][START_REF] Hanin | Which neural net architectures give rise to exploding and vanishing gradients?[END_REF]. Without proper adaptation or training, they show poor performance. To cope with this issue, highway networks [START_REF] Rupesh | Training very deep networks[END_REF] and residual networks [START_REF] He | Deep residual learning for image recognition[END_REF] were introduced. Their many successive layers behaves either as a dense layer or as the identity function, allowing the gradient information to propagate trough the successive layers.

Introduction

We compare the benefits of preconditioned Langevin algorithms [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] for various architectures and depths of neural networks and we proceed to side-to-side comparison of Langevin algorithms with their respective non-Langevin counterparts. The purpose of our experiments is to compare different methods on the same model architecture, not to achieve state-of-the-art results. For shallow networks, there is no benefit in using Langevin algorithms as it only adds noise to the gradient descent and brings a less accurate estimation of the minimum. However, we observe that the deeper the network is, the greater are the gains provided by Langevin algorithms.

Since the most important non-linearities of the network are contained in the deepest layers, we introduce a new optimization method that we call Layer Langevin algorithm, which consists in training the network by adding Langevin noise only to the training of some layers and not to the other layers. In particular, we choose the Langevin layers to be the k first (deepest) layers for some integer k. We then highlight the possibilities of training acceleration using Langevin and Layer Langevin methods on deep residual networks [START_REF] He | Deep residual learning for image recognition[END_REF] and dense convolutional networks [START_REF] Huang | Densely connected convolutional networks[END_REF] for image classification.

Our code for the numerical experiments is available at https://github.com/Bras-P/deep-layer-langevin.

It includes in particular ready-to-use Langevin optimizers and Layer Langevin optimizers as instances of the TensorFlow Optimizer base class and a demonstration notebook.

Very deep neural networks

Training of very deep neural networks is a significantly more challenging task than for shallow networks [GB10, DPG + 14]. Let us write the output of a neural network with K layers and with weights θ = (θ 1 , . . . , θ K ) as

ψ θ (x) = φ K θ K • • • • • φ 1 θ 1 (x), (6.2.1)
where φ 1 , . . . , φ K are activation function and where

φ k θ k : x → φ k (θ k • x) at every unit. Denoting Φ k (x) := φ k θ k • • • • • φ 1 θ 1 (x) (6.2.2)
for 1 ≤ k ≤ K and Φ 0 (x) := x, then the gradient reads for 1 ≤ k ≤ K:

∇ θ k ψ θ (x) = (∇ θ K φ θ K • Φ K-1 (x)) • • • • • (∇ θ k φ θ k • Φ k-1 (x)) . (6.2.3)
Thus heuristically, the deeper the layer is, the more the gradient with respect to the parameters of this layer has annealing points, since more factors appear in (6.2.3), hinting that deep layers show more non-linearities and local traps.

Langevin algorithms for the training of deep neural networks 6.3.1 Experimental setting

In our experiments we use the following datasets. The MNIST dataset [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF] is composed of 28 × 28 grayscale images of handwritten digits (from 0 to 9). 60.000 images are used for training and 10.000 images are used for test. The CIFAR-10 and the CIFAR-100 datasets [START_REF] Krizhevsky | Learning multiple layers of features from tiny images[END_REF] consist in RGB images of size 32 × 32 belonging to 10 and 100 different classes respectively. For both datasets 50.000 images are used for training and 10.000 images are used for test.

The neural networks are trained using preconditioned Langevin algorithms with per-dimension adaptive stepsize [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] with different choices of preconditioner. That is, for a preconditioner rule (P n ) the Langevin update reads

g n+1 = ∇ θ V (θ n ; D n+1 ) (6.3.1) θ n+1 = θ n -γ n+1 P n+1 • g n+1 + σ √ γ n+1 N (0, P n+1 ), (6.3.2)
where σ ∈ (0, ∞) controls the amount of injected noise, (γ n ) is the non-increasing learning rate sequence, V denotes the objective function and where ∇ θ V (θ n ; D n ) stands for the mean gradient computed on a subset D n of the dataset. The corresponding preconditioned non-Langevin algorithm follows the same update as in (6.3.2) without Gaussian noise. In our experiments we use the RMSprop [DHS11, LCCC16], the Adam [KB15] and the Adadelta [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] preconditioners and we call the Langevin version of these algorithms as L-RMSprop, L-Adam and L-Adadelta respectively. The preconditioner rules are given in Algorithms 4, 5, 6 respectively. Note that depending on the algorithm version, in the update (6.3.2) the gradient g n+1 can be replaced by an averaged gradient over the past iterations as this in the case in Adam (Algorithm 5) i.e. momentum gradient is used. While comparing some preconditioned method with its Langevin counterpart, we ensure that both training procedures start with the same initial weights.

Algorithm 4 RMSprop update

Parameters: α, λ > 0 MS n+1 = α MS n +(1 -α)g n+1 ⊙ g n+1 P n+1 = diag 1 ⊘ λ1 + √ MS n+1 θ n+1 = θ n -γ n+1 P n+1 • g n+1 Algorithm 5 Adam update Parameters: β 1 , β 2 , λ > 0 M n+1 = β 1 M n + (1 -β 1 )g n+1 MS n+1 = β 2 MS n +(1 -β 2 )g n+1 ⊙ g n+1 M n+1 = M n+1 /(1 -β n+1 1 ) MS n+1 = MS n+1 /(1 -β n+1 2 ) P n+1 = diag 1 ⊘ λ1 + MS n+1 θ n+1 = θ n -γ n+1 P n+1 • M n+1 .

Plain and convolutional networks

We first train fully connected feedforward neural networks on the MNIST dataset. The networks are composed of 3, 20, 30 and 40 hidden dense layers respectively with 64 units each and with Algorithm 6 Adadelta update Parameters:

β 1 , β 2 , λ > 0 MS n+1 = β 1 MS n +(1 -β 1 )g n+1 ⊙ g n+1 P n+1 = diag M S n + λ1 ⊘ λ1 + MS n θ n+1 = θ n -γ n+1 P n+1 • g n+1 . M S n+1 = β 2 MS n +(1 -β 2 )(θ n+1 -θ n ) ⊙ (θ n+1 -θ n ).
ReLU activation, followed by one dense output layer. The results are given in Figure 6.1. We observe that for shallow neural networks, Langevin algorithms do not outperform their respective non-Langevin counterparts; they add noise to the gradient descent thus giving a less accurate estimate of the minimum value. In particular and as noted in the footnote in [START_REF] Marceau-Caron | Natural Langevin Dynamics for Neural Networks[END_REF], we could not reproduce the good results from [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] for plain networks with two hidden layers. However, the deeper the network is, the greater the gains induced by Langevin algorithms compared with their respective non-Langevin counterparts are. We also display the value of the loss function on the training set in order to highlight that the better performances of the Langevin algorithms are not due to some overfitting effect. Langevin algorithms indeed show improvements on 20-layer deep networks; beyond 30-layer deep networks, the gains are significant. The training of 40-layer deep networks with non-Langevin algorithms may run into the vanishing gradient problem, whereas such problem is avoided by Langevin algorithms. In the latter case of very deep training, preconditioned Langevin algorithms not only add noise preventing the vanishing of the gradient, they also help starting up the training in the right directions. To obtain better results with Langevin algorithms, we recommend using a small coefficient σ, empirically ranging from 1 e -3 to 5 e -5.

We then perform simulations in a similar setup on convolutional architectures that are more adapted to image recognition [START_REF] Jarrett | What is the best multi-stage architecture for object recognition?[END_REF] followed by a large number of hidden dense layers. More specifically, we train neural networks composed of two convolutional layers with 4×4 kernel size and 32 channels for each; 2×2 max-pooling is used after each convolutional layer. These layers are followed by respectively 10 and 30 hidden dense layers with 64 units each and by one dense output layer. Since the images in the CIFAR-10 dataset do not have a good resolution, we cannot expect a very high accuracy on the test set. Instead, we focus on comparing different algorithms on the same model architecture. The results are given in Figure 6.2 and we make similar observations: Langevin algorithms show improvements with 10 hidden dense layers and for 30 dense layers, non-Langevin algorithms run into vanishing gradient issues which is not the case for Langevin algorithms.

Highway networks

We now perform the same simulations on Highway networks, in a setting very similar to [START_REF] Rupesh | Training very deep networks[END_REF]. Comparably to residual networks, the output of a highway layer is a convex combination of the output of a dense layer and the output of a identity layer; the parameter controlling the convex combination is itself trainable. For a layer with weights (θ D , θ T ), the output reads

y = T θ T (x) • D θ D (x) + (1 -T θ T (x)) • x, (6.3.3)
where T and D are dense layers and where T has sigmoid output. We give the implementation of a Highway layer as a subclass of the base TensorFlow class tf.keras. layers.Layer. The method build initializes the weights of the layer when the layer is called for the first time; each weight is defined through the method add_weight. The method call defines the output of the layer depending on the inputs and its weights; we need to resort We observe that Langevin algorithms become effectively faster than non-Langevin algorithms only from a larger depth than for plain networks. In Figure 6.3 we plot the results for the training of a network composed of 80 dense hidden layers with 64 units each and ReLU activation on the CIFAR-10 dataset, showing the possibilities of acceleration through Langevin algorithms, even in a residual (highway) architecture.

Layer Langevin algorithm

We introduce a new Langevin algorithm for stochastic optimization of deep neural networks that we call Layer Langevin algorithm. Choosing a preconditioner rule P , some weights are updated following the Langevin rule while the other weights are updated following the non-Langevin rule. Denoting θ (i) n the ith weight at step n, we have for every i:

θ (i) n+1 = θ (i) n -γ n+1 [P n+1 • g n+1 ] (i) + 1 i∈J σ √ γ n+1 N (0, P n+1 ) (i) , (6.4.1)
where J is a subset of weight indices and where P n denotes the preconditioner. To simplify the choice of J , we choose J as the subset of indexes of weights belonging to some layers. However, a finer control over the subset J remains possible. To implement this method in practice, we simply assign before the training an attribute equals to 1 i∈J to every trainable variable of the network.

We compare the performances of Layer Langevin algorithms with the Adam preconditioner for different choices of the subset of layers. The results are given in Figure 6.4 for the training of a dense network with 30 hidden dense layers on the MNIST dataset in a setting similar to Figure 6.1. For some optimizer Name, we denote LL-Name p% the corresponding Layer Langevin algorithm where the subset J is the first p% layers of the network. We observe that we obtain significant gains in comparison with the vanilla Langevin algorithm and that the best performances are obtained when choosing the subset J as being the first ℓ layers for some ℓ ∈ N, in particular all the layers of the network except the few last ones. The architecture is composed of 5 convolution blocks; inside each block the number of channels is kept constant. At each transition between two blocks, the height and the width of the image are both divided by 2 while the number of channels is multiplied by 2 (except for the last block). Two hidden fully connected layers are used at the end.

Application to deep architectures for image recognition

We now test the Layer Langevin algorithm to speed up the training of neural networks with very deep architectures that are popular in image recognition. Many advances in supervised learning were made possible using very deep neural networks, which are able to tackle much more difficult problems than shallow ones. In particular, let us focus on very deep convolutional networks for image recognition. VGG (Visual Geometry Group) networks were introduced in [SZ15] and then became a standard for deep convolutional neural networks. The VGG architecture consists in a succession of 2D convolutional layers with kernel size 3 × 3 and with ReLU activation; a batch normalization layer is applied before each convolutional layer. The dimensions of the input image are gradually reduced using 2×2 pooling layers while the number of channels is increased. At the end of the network, dense layers are applied for class prediction. An illustration of an example of VGG architecture is given in Figure 6.5.

However, as this is the case for fully connected neural networks, very deep convolutional networks are considerably more difficult to train. To cope with this issue, residual networks (ResNets) were introduced in [START_REF] He | Deep residual learning for image recognition[END_REF]. In this architecture, in order to propagate the gradient information through the many layers a residual learning is implemented every few stacked layers. More precisely, ResNets are composed of a succession of blocks as described in Figure 6.6. Each of these blocks includes a shortcut connection and operates in part as a simple identity layer. Contrary to a highway connection, the coefficient of the shortcut connection is 1 and is nontrainable. Similarly to VGG nets, the size of the image is gradually reduced using convolutional layers with 2 × 2 strides, see Figure 6.7. An example of ResNet architecture is given in Figure 6.8. Residual networks allow substantially deeper architectures and gain accuracy from increased depth [START_REF] He | Deep residual learning for image recognition[END_REF].

An implementation of residual networks using the TensorFlow functional API is given in Listing 6.1. The (main) arguments are:

• block_layers: list of the number of blocks between two size reductions. For example,

[2,3,3] yields the architecture in Figure 6.8.

• filters: initial number of channels which is multiplied by 2 at every size reduction.

• mode: either 'resent' or 'vgg', in order to instantiate either the ResNet of the VGG architecture.

An implementation of the VGG architecture can be directly obtained from Listing 6.1 by skipping the shortcut connections in identity_block and convolutional_block. , are composed of convolutional layers; within every DenseNet blocks, each block is connected to all the following layers inside the block, as described in Figure 6.9. Contrary to residual networks where simply adding the features implies some loss of information, the layers are not connected by summation but by concatenation, i.e. for y 0 , . . . , y ℓ the outputs of the previous layers and H ℓ the ℓ layer in the block which is composed of 2D convolution, activation and batch normalization, the output of the ℓ th layer is H ℓ ([x 0 , . . . , x ℓ ]) where [x 0 , . . . , x ℓ ] denotes the concatenation of the (same sized) multichannel images x 0 , . . . , x ℓ , whereas the output for a resnet architecture is H ℓ (x ℓ ) + x ℓ-1 . This architecture allowing feature reuse and improves the flow of information throughout the network , which helps training of deeper architectures. Likewise VGG and residual networks, the different blocks are concatenated using a convolutional layer and a pooling layer so that the size of the image is progressively reduced. Dense convolutional networks obtained significant improvements over previous architectures while also reducing the number of parameters [START_REF] Huang | Densely connected convolutional networks[END_REF]. (5 e -3 for Adadelta) for epochs 1 to 40 γ n is divided by 10 and σ is set to 0 beyond.

We train on the CIFAR-10 dataset a ResNet architecture composed of 2 blocks with 5 residual layers each; each block is followed by a size reduction layer. This architecture is given as ResNet-20 in [HZRS16, Section 4.2]. Similarly, we train a DenseNet architecture with depth L = 22 and growth order k = 5 [START_REF] Huang | Densely connected convolutional networks[END_REF] on the CIFAR-10 dataset. We apply usual data augmentation to both CIFAR-10 and CIFAR-100 datasets [LXG + 15, HZRS16]: 4 pixels are padded on each side and a 32 × 32 crop is randomly sampled from the padded image or its horizontal flip. The results are given in figure 6.10 and tables 6.1 and 6.2. Experiments show that Layer Langevin algorithms (in this case LL-Adam 30%) yield improvements in comparison with non-Langevin methods, even on residual architectures adapted to very deep learning. The train loss is also plotted, showing that the better performances of Layer Langevin is not only due to some overfitting effect.

Chapter 7

Langevin algorithms for Markovian Neural Networks and Deep Stochastic control

The results in this chapter have been presented at the Machine Learning conference International Joint Conference on Neural Networks IJCNN 2023 and are published in the conference proceedings as a joint work with Gilles Pagès [START_REF] Bras | Langevin algorithms for markovian neural networks and deep stochastic control[END_REF].

Abstract

Stochastic Gradient Descent Langevin Dynamics (SGLD) algorithms, which add noise to the classic gradient descent, are known to improve the training of neural networks in some cases where the neural network is very deep. In this paper we study the possibilities of training acceleration for the numerical resolution of stochastic control problems through gradient descent, where the control is parametrized by a neural network. If the control is applied at many discretization times then solving the stochastic control problem reduces to minimizing the loss of a very deep neural network. We numerically show that Langevin algorithms improve the training on various stochastic control problems like hedging and resource management, and for different choices of gradient descent methods.

Keywords-Langevin algorithm, SGLD, Markovian neural network, Stochastic control, Deep neural network, Stochastic optimization.

Introduction

Stochastic Optimal Control (SOC), which consists in optimizing a functional of a trajectory of a controlled Stochastic Differential Equation (SDE) has applications in a wide range of problems: management of resources, queuing systems, epidemic and population processes, pricing of financial derivatives, portfolio allocation... In comparison with classic optimal control, SOC models include a random noise with known probability distribution that affects the evolution or the observation of the system. SOC also aims at managing the risk induced by this noise. SOC problems are usually solved using specific strategies, such as Forward-Backward SDEs (FBSDEs) [START_REF] Peng | Fully coupled forward-backward stochastic differential equations and applications to optimal control[END_REF], or by solving Hamilton-Jacobi-Bellman (HJB) optimality conditions [START_REF] Bellman | Dynamic programming[END_REF] through partial differential equations methods using appropriate numerical schemes or by stochastic dynamic programming [START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF]. Such problems can also be solved using Neural Networks calibrated by SGD techniques [GM05, HE16, WLP + 19, CL21, BHLP22].

More specifically, in this article we consider the numerical resolution of a SOC problem where the control is parametrized by a neural network calibrated by gradient descent. This method implies to compute the path-wise derivatives along the trajectory of the SDE of the objective function with respect to the parameters of the neural network, as introduced in [START_REF] Giles | Smoking adjoints: fast evaluation of Greeks in Monte Carlo calculations[END_REF][START_REF] Michael | Monte Carlo evaluation of sensitivities in computational finance[END_REF]. Stochastic gradient descent is a very general approach that can be applied to a wide range of problems and which does not need to be specifically adapted to each problem under consideration. Moreover, SGD scales very efficiently to high-dimensional problems, in contrast with HJB-based methods, and has proved its efficiency on highly non-convex problems [START_REF] Dauphin | Equilibrated adaptive learning rates for non-convex optimization[END_REF].

However, if the neural control is applied at many time steps as it is the case for the Euler-Maruyama scheme where the (discrete) control is taken as an approximation of a continuous control, then the SOC problem reads as the optimization of a very deep neural network, which is roughly as deep as the number of instants at which the control can be applied (see Figures 7.1 and 7.2). Very deep neural networks are known to be considerably more difficult to train [GB10, DPG + 14] and may run into vanishing gradient problems [START_REF] Hochreiter | Investigations on dynamic Neural Networks[END_REF][START_REF] Hanin | Which neural net architectures give rise to exploding and vanishing gradients?[END_REF]. Indeed, the deeper the neural network is, the more non-linear it is, thus increasing the number of local traps for the gradient descent such as local (but not global) and saddle points. In image analysis where very deep convolutional neural networks are commonly used, residual [START_REF] He | Deep residual learning for image recognition[END_REF] and convolutional dense [START_REF] Huang | Densely connected convolutional networks[END_REF] networks were introduced to deal with this issue. These networks are based on architectures with residual connections to propagate the gradient information through the numerous successive layers.

As it comes to deep SOC, we cannot freely change the structure of the neural network since it is fixed by the equations defining the SOC problem and therefore we cannot directly use residual connections. We can only freely choose the structure of the neural network returning the control, for which a few layers is often enough (see for example [START_REF] Buehler | Deep hedging[END_REF][START_REF] Bachouch | Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications[END_REF]).

A way to improve the learning is to replace SGD algorithms by Stochastic Gradient Langevin Dynamics (SGLD) algorithms. Such optimizers add an exogenous white noise to the gradient descent, providing regularization and allowing to escape from traps. It has indeed been observed that adding noise improves the learning for very deep neural networks [NVL + 15, Ani19, GMDB16, SLH + 19]. Moreover, Chapter 6 compares side-by-side Langevin with non-Langevin algorithms on networks with increasing depth and shows that for shallow neural networks, Langevin algorithms do nothing else than adding noise to the gradient descent, however the deeper the network is, the greater the gains provided by Langevin algorithms are. We also refer to Chapter 5.

In the present article we study the performances of Langevin optimizers on SOC problems where the number of discretization times where the control is applied is large enough. We use the preconditioned versions of SGD and SGLD [START_REF] Li | Preconditioned stochastic gradient langevin dynamics for deep neural networks[END_REF] for various choices of preconditioners. We compare side-by-side Langevin and non-Langevin algorithms and we show that Langevin optimizers can significantly improve the training procedure on various problems: fishing quotas [START_REF] Mathieu Laurière | Performance of a Markovian Neural Network versus dynamic programming on a fishing control problem[END_REF], deep hedging [START_REF] Buehler | Deep hedging[END_REF], oil drilling and resource management [START_REF] Gaïgi | Optimal risk management problem of natural resources: application to oil drilling[END_REF]. We mainly consider two different approaches for numerical resolution of SOCs. In the first approach, the control is a single neural network which is applied to every time step and which may depend on the running time t (see Figure 7.1). This approach leads to a model with fewer trainable parameters, which is critical in some data-driven financial applications where the amount of data is limited, and which is more able to capture the specific Markovian features of the problem. In the second approach, a different neural network is used for each control time (see Figure 7.2). This last setting is also suitable for applying the Layer Langevin algorithm, which is a variant of the Langevin algorithm introduced in Chapter 6 and which proved to be more adapted to the training of very deep neural networks than the Langevin algorithm itself.

We observe that the gains of Langevin algorithms depend on the preconditioner however. While the Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] and the Adadelta [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] algorithms can be substantially accelerated by Langevin training, the gains are more limited or sometimes null for RMSprop [START_REF] Tieleman | Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude[END_REF]. The code for the numerical experiments is available at https://github.com/Bras-P/langevin-for-stochastic-control.

It includes in particular ready-to-use Langevin optimizers and Layer Langevin optimizers as instances of the TensorFlow Optimizer base class, a framework for algorithm comparison in a SOC setting with GPU support and a demonstration notebook.

In the following, on top of the notations defined from page 1, we consider multivariate (F t )-Brownian motions W and B defined on some filtered probability space (Ω, F, (F t ) t∈[0,T ] ), P).

Stochastic control through gradient descent

Stochastic optimal control

We consider the following SOC problem in continuous time:

min u J(u) := E T 0 G(X t )dt + F (X T ) , (7.2.1) dX t = b(X t , u t )dt + σ(X t , u t )dW t , t ∈ [0, T ] (7.2.2)
where b : R

d 1 × R d 3 → R d 1 , σ : R d 1 × R d 3 → M d 1 ,d 2 (R), W is a R d 2 -valued Brownian motion and u is a R d 3 -valued continuous (F t )-adapted process, T > 0, G : [0, T ]×R d 1 → R and F : R d 1 → R.
We first approximate the continuous SDE (X t ) t∈[0,T ] with its Euler-Maruyama scheme and the control u t with a discrete-time control. For N ∈ N being the number of discretization times, we consider the regular subdivision of [0, T ]:

t k := kT /N, k ∈ {0, . . . , N }, h := T /N (7.2.3)
and we approximate the control applied at times t 0 , . . . , t N -1 as the output of a single neural network depending on t, or as the output of N neural networks, one for each discretization instant t k :

u t k = ūθ (t k , X t k , H t k ) or u t k = ūθ k (X t k , H t k ) (7.2.4)
where

H t := t 0 G(X s )ds,
where ūθ is a neural function with finite-dimensional parameter θ ∈ R d . Indeed, since (7.2.2) defines a Markovian process, we can assume that u t depends only on t, X t and H t instead of t and (X s ) s∈[0,t] .

The SOC problem (7.2.1) is numerically approximated by:

min θ J(ū θ ) := E N -1 k=0 (t k+1 -t k )G( Xθ t k+1 ) + F ( Xθ t N ) , (7.2.5) Xθ t k+1 = Xθ t k + (t k+1 -t k )b Xθ t k , ūk,θ ( Xθ t k , Hθ t k ) + t k+1 -t k σ Xθ t k , ūk,θ ( Xθ t k , Hθ t k ) ξ k+1 , (7.2.6) Hθ t k+1 = Hθ t k + (t k+1 -t k )G( Xθ t k+1 ), ξ k ∼ i.i.d. N (0, I d 2 )
where θ ∈ R d and ūk,θ = ūθ (t k , •) in the first case of (7.2.4) and θ = (θ 0 , . . . , θ N -1 ) ∈ (R d ) N and ūk,θ = ūθ k in the second case of (7.2.4).

For every θ, ∇ θ J can be computed by automatic differentiation as the gradient w.r.t. to θ is tracked all along the trajectory through the recursive relation (7.2.6) [START_REF] Giles | Smoking adjoints: fast evaluation of Greeks in Monte Carlo calculations[END_REF][START_REF] Michael | Monte Carlo evaluation of sensitivities in computational finance[END_REF]. Then the SGD algorithm reads

θ n+1 = θ n -γ n+1 1 n batch n batch i=1 ∇ θ J(ū θn , (ξ i,n+1 k ) 1≤k≤N ) =: θ n -γ n+1 g n+1 (7.2.7)
where (ξ i,n k ) 1≤k≤N,1≤i≤n batch ,n∈N is an array of i.i.d. random vectors N (0, I d 2 )-distributed, (γ n ) n∈N is a non-increasing positive step sequence and where the dependence of J in (ξ i,n k ) is made explicit.

If the number of Euler-Maruyama steps N is large, then the optimization problem in (7.2.5) consists in the training of a very deep neural network that can be difficult to train directly (see the Introduction). Both cases are illustrated in Figures 7.1 and 7.2.

Preconditioned stochastic gradient Langevin dynamics

We consider preconditioned stochastic gradient algorithms i.e. for (P n ) a preconditioner rule the update reads

θ n+1 = θ n -γ n+1 P n+1 • g n+1 (7.2.8)
where g n+1 is defined in (7.2.7). We use the Adam [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF], the RMSprop [TH12] and the Adadelta [START_REF] Matthew | ADADELTA: An Adaptive Learning Rate Method[END_REF] preconditioners, which are detailed in Algorithms 7, 8 and 9 respectively. For some algorithm name, the corresponding Langevin algorithm denoted L-name reads

θ n+1 = θ n -γ n+1 P n+1 • g n+1 + σ n+1 √ γ n+1 N (0, P n+1 ) (7.2.9)
where (σ n ) is a constant or non-decreasing sequence controlling the amount of injected noise.

Algorithm 7 Adam update Parameters:

β 1 , β 2 , λ > 0 M n+1 = β 1 M n + (1 -β 1 )g n+1 MS n+1 = β 2 MS n +(1 -β 2 )g n+1 ⊙ g n+1 M n+1 = M n+1 /(1 -β n+1 1 ) MS n+1 = MS n+1 /(1 -β n+1 2 ) P n+1 = diag 1 ⊘ λ1 + MS n+1 θ n+1 = θ n -γ n+1 P n+1 • M n+1 . Algorithm 8 RMSprop update Parameters: α, λ > 0 MS n+1 = α MS n +(1 -α)g n+1 ⊙ g n+1 P n+1 = diag 1 ⊘ λ1 + √ MS n+1 θ n+1 = θ n -γ n+1 P n+1 • g n+1
The Layer Langevin algorithm, introduced in Chapter 6 Section 6.4 consists in updating with Langevin noise only some layers of the network. It relies on the heuristic that for a deep neural network, the non-linearities of the network are mostly contained in the deepest layers and adds Langevin noise to these layers only.

Choosing a preconditioner rule P called name, the Layer Langevin algorithm denoted LLname reads

θ (i) n+1 = θ (i) n -γ n+1 [P n+1 • g n+1 ] (i) + 1 i∈J σ n+1 √ γ n+1 N (0, P n+1 ) (i) , (7.2.10)
Langevin algorithms to be suitable for the fine tuning, but since the first step of the training still consists in training a deep neural network, we analyse the benefits of Langevin algorithms for this first step. Unless stated otherwise, the batch size is set to n batch = 512 i.e. stochastic gradient iterations are performed by averaging the gradient over 512 random trajectories. In the plot, each epoch consists in 5 batches i.e. the average loss J(u θ ) is plotted every 5 iterations of the stochastic gradient. After each epoch, J(u θ ) is estimated over 25 × 512 trajectories and 95% confidence intervals are indicated, although for some plots the intervals are too small to be visible. While comparing some algorithm with its Langevin or Layer Langevin counterpart, we ensure that both training procedures start with the same initial weights.

Fishing quotas

We consider the fishing quota problem introduced in [START_REF] Mathieu Laurière | Performance of a Markovian Neural Network versus dynamic programming on a fishing control problem[END_REF]. Let X t ∈ R d 1 be the fish biomass for every fish species; we wish to keep it close to an ideal state X t ∈ R d 1 . The dynamics of X are given by

dX t = X t * ((r -u t -κX t )dt + ηdW t ) , t ∈ [0, T ], (7.3.1)
where r ∈ R d 1 is the growth rate for each species, u t ∈ R d 1 is the controlled fishing (with ) is the interaction matrix between the fish species,

d 3 = d 1 ), κ ∈ M d 1 ,d 1 (R
η ∈ M d 1 ,d 2 (R), W is a R d 2 -valued Brownian motion. The control u is constrained to take its values in [u m , u M ] d 1 .
The objective is

J(u) = E T 0 (|X t -X t | 2 -⟨α, u t ⟩)dt + β[u] 0,T , (7.3.2) where α ∈ R d 1 , β ∈ R + , [u] 0,T denotes the quadratic variation of u on [0, T ].
The term ⟨α, u⟩ penalizes small fishing quotas while the term β[u] 0,T penalizes too many daily changes.

In the experiments, following [START_REF] Mathieu Laurière | Performance of a Markovian Neural Network versus dynamic programming on a fishing control problem[END_REF] we choose

d 1 = d 2 = 5, T = 1, X ≡ 1, r = 2 * 1, η = 0.1 * I d 1 , α = 0.01 * 1, β = 0.1, u m = 0.1, u M = 1 (7.3.3) and κ =        1.2 -0.1 0 0 -0.1 0.2 1.2 0 0 -0.1 0 0.2 1.2 -0.1 0 0 0 0.1 1.2 0 0.1 0.1 0 0 1.2        . (7.3.4)
The initial state X 0 is randomly generated following

N 1, (1/2)I d 1 clipped to [0.2, 2] d 1 . The quadratic variation [u] 0,T is approximated in the discretized setting by [u] 0,T ≃ N -1 k=0 |u t k+1 -u t k | 2 . (7.3.5)
Each control u θ is given by a feedforward neural network with two hidden layers with 32 units each and with ReLU activation while the output layer has sigmoid activation in order to fulfil the constraint on u. An example of controlled trajectory is plotted in Figure 7.3. The results are given in Figure 7.4 for the Adam optimizer with an increasing number of Euler-Maruyama steps and with one single control, in Figure 7.5 for the RMSprop and L-Adadelta optimizers and with one single control and in Figure 7.6 for the training with multiple neural networks.

Deep hedging

We consider the problem of hedging portfolio of derivatives as a SOC problem as in [START_REF] Buehler | Deep hedging[END_REF]. We aim to replicate a F T -measurable payoff Z defined on some portfolio S t ∈ R d 1 by trading (at least some of) the assets contained in S t at times (t k ). The control is given by u t ∈ R d 1 representing the amount held for each asset. The objective is

J(u) = ν -Z + N -1 k=0 ⟨u t k , S t k+1 -S t k ⟩ - N k=0 ⟨c tr , S t k * |u t k -u t k-1 |⟩ (7.4.1)
where ν :

L 1 (Ω) → R is a convex risk measure (see [BGTW19, Definition 3.1]), c tr ∈ R d 1
represents proportional transaction costs and we fix u t -1 = u t N = 0, implying full liquidation in T . We furthermore assume that ν can be written as

ν(X) = inf w∈R (w + E[ℓ(-X -w)]) (7.4.2)
where the loss function ℓ : R → R is continuous, non-decreasing and convex. This is the case in particular for the entropic risk measure where ℓ(x) = -exp(-λx) and the conditional value at risk measure where ℓ(x) = (1 -α) -1 max(x, 0). Then (7.4.1) can be rewritten as

inf u,w J(u, w) := E w + ℓ Z - N -1 k=0 ⟨u t k , S t k+1 -S t k ⟩ + N k=0 ⟨c tr , S t k * |u t k -u t k-1 |⟩ -w .
(7.4.3) In the numerical experiments, we analyse the problem of hedging in a Heston model as described in [BGTW19, Section 5]. For even d 1 , we consider d ′ 1 := d 1 /2 independent Heston models where the price and volatility processes are described by the following SDEs for 1

≤ i ≤ d ′ 1 : dS 1,i t = V i t S 1,i t dB i t , S 1,i 0 = s i 0 , (7.4.4
)

dV i t = a i (b i -V i t )dt + η i V i t dW i t , V i 0 = v i 0 , (7.4.5)
where a, b, η, s 0 , v i ∈ (R + ) d ′ 1 and for each 1 ≤ i ≤ d ′ 1 , B i and W i are standard Brownian motions with correlation ρ i ∈ [-1, 1]. The volatility V itself is not tradable directly but only through options on variance modelled by the following variance swap:

S 2,i t := E T 0 V i s ds F t = t 0 V i s ds + L i (t, V i t ), (7.4.6) L i (t, v) := v -b i a i 1 -e a i (T -t) + b i (T -t). (7.4.7)
The payoff is given by

Z = d ′ 1 i=1 S 1,i T -K i + where K ∈ (R + ) d ′ 1 .
We consider the convex risk measure associated to the value-at-risk i.e. associated to the loss function

ℓ(x) = 1 1 -α max(x, 0).
In the experiments we choose

d ′ 1 = 5, T = 1, a = 1, b = 0.04 * 1, η = 2 * 1, ρ = -0.
7 * 1, α = 0.9, (7.4.8) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 for Adam (resp. Adadelta) for epochs 0 to 80 and γ n is divided by 10 and σ n is set to 0 beyond.

1 1.1 0 1 2 t S 1 t V t u 1 t /2 u 2 t /10
s 0 = K = 1, v 0 = 0.1 * 1, c tr = 5 e -4 * 1. (7.4.9)
Each control u θ is given by a feedforward neural network with two hidden layers with 32 units each and with ReLU activation while the output layer has ReLU activation too in order to forbid short-selling. As recommended in [START_REF] Buehler | Deep hedging[END_REF], since transaction costs are involved the control u θ at time t k is a function of log(S 1 t k ), V t k and u t k-1 . An example of controlled trajectory showing only one of the five Heston models is plotted in Figure 7.7.

The results are given in Figure 7.8 for the comparison of Langevin and non-Langevin algorithms with a single control and in Figure 7.9 for the training with multiple controls.

Resource management

We consider the control problem in the management of natural resources applied to oil drilling introduced in [GKL18] and extended in [START_REF] Gaïgi | Optimal risk management problem of natural resources: application to oil drilling[END_REF]. The objective is for an oil driller, to balance the costs of extraction, storage in a volatile energy market. The oil price P t ∈ R is assumed to be a Black-Scholes process: (5 e -1) and σ n = 2 e -3 (5 e -3) for Adam and RMSprop (resp. Adadelta) for epochs 0 to 180 and γ n is divided by 10 and σ n is set to 0 beyond. The control is given by q t = (q v t , q s t , q v,s t ) ∈ R 3 where q v t is the quantity of extracted oil immediately sold on the market per time unit, q s t is the quantity of extracted oil that is stored per time unit, q v,s t is the quantity of stored oil that is sold per time unit. The cumulated quantities of extracted and stored oil at time t are respectively given by

dP t = µP t dt + ηP t dW t . ( 7
E t = t 0 (q v r + q s r )dr, S t = t 0 (q s r -q v,s r )dr. (7.5.2)
The extraction and storage prices are respectively given by c e (E t ) = exp (ξ e E t ) , c s (S t ) = exp (ξ s S t ) -1. (7.5.3)

The constraints on the control are the following:

q v t , q s t , q v,s t ≥ 0, q v,s t ≤ q S , q v t + q s t ≤ K 0 , 0 ≤ S t ≤ Q S , (7.5.4)
where q S , K 0 and Q S are operational bounds. The objective is

J(q) = -E T 0 e -ρr U q v r P r + q v,s r (1 -ε)P r -(q v r + q s r )c e (E r ) -c s (S r ) dr , ( 7.5.5)
where U : R → R is the utility function.

In the experiments we take

T = 1, µ = 0.01, η = 0.2, ρ = 0.01, ε = 0, K 0 = 5, ξ e = 1 e -2, ξ s = 5 e -3, q S = 10, P 0 = 1, U (x) = x. (7.5.6)
The control q t is given by a feedforward neural network with two hidden layers with 32 units and with ReLU activation while the output layer has several ReLU activations such that the constraints on q (7.5.4) are fulfilled1 . An example of controlled trajectory is given in Figure 7.10.

The results are given in Figure 7.11 for the comparison of Langevin and non-Langevin algorithms with a single control. We do not display the results for the training with multiple controls however as we could not obtain satisfying results neither with Langevin nor-with non-Langevin methods. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 (2 e -3, 5 e -3) for Adam (resp. RMSprop, Adadelta) for epochs 0 to 60 (resp. 80, 80) and γ is divided by 10 and σ is set to 0 beyond.

1 1.1 0 1 2 t P t E t S t

Comments on the numerical experiments

We observe that in many cases and in various SOC problems, Langevin and Layer Langevin algorithms show improvement when compared with their respective non-Langevin counterparts, provided that N is large enough, which is remarkable for randomized algorithms. Langevin algorithms converge faster and/or toward a lower loss value. This is particularly visible for the Adadelta method. The gains are limited in some cases (see Figures 7.5 and 7.11 for RMSprop) but still the optimization procedure is improved.

The gains for the L-RMSprop algorithm remain limited however. In particular, we did not observe any significant improvement for fishing SOC with multiple controls, for deep hedging SOC with a single control and for oil drilling SOC. We do not have explanation for this fact.

The gains brought by Langevin algorithm increase with the depth of the network as shown in Figures 7.4 and 7.8. However, contrary to Chapter 6, we did not observe overwhelming gains as N becomes very large. We believe that this is due (in part) to the particular structure of the deep SOC problem where the same control is repeated all along the trajectory.

As for SOC with multiple controls, we observe that Layer Langevin algorithms with a small number of Langevin layers (10%-30%) generally outperforms Vanilla Langevin methods while Vanilla Langevin may bring limited gains or be less efficient than the standard non-Langevin methods, see Figures 7.6 and 7.9 for Adam.

Part III

Numerical simulation of stochastic processes 8.1 Introduction

The convergence properties of Euler-Maruyama schemes to approximate the solution of a Stochastic Differential Equation (SDE) have been extensively studied, in particular for L p distances. However, the literature seems to lack some results about the convergence in total variation in small time. More specifically, in this paper we consider the two following SDEs in R d starting at the same point:

X x 0 = x ∈ R d , dX x t = b 1 (t, X x t )dt + σ 1 (t, X x t )dW t , Y x 0 = x, dY x t = b 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t ,
where W is a Brownian motion. We generally assume that for i = 1, 2, b i is Lipschitz continuous and that σ i is elliptic, bounded and Lipschitz continuous, but we do not assume that b i is bounded. Our objective is to give bounds of the total variation distance between the law of X x t and the law of Y x t , denoted d TV (X x t , Y x t ), as t → 0. In particular, we apply our results to the case where Y x = Xx is the one-step Euler-Maruyama scheme associated to the SDE X, given by dY x t = b 1 (0, x)dt + σ 1 (0, x)dW t . Such bounds are well known for L p distances and their associated Wasserstein distances and are known to be of order t as t → 0. Yet the literature seems to lack results as it comes to d TV .

If

σ 1 = σ 2 is constant, then it is classical background that d TV (X x t , Y x t
) is of order t, using a Girsanov change of measure (see for example [PP23, Proposition 4.1]) but this strategy cannot be applied to non-constant σ. The difficulty of the total variation distance in small time is the following: considering its representation formula and comparing it with the L 1 -Wasserstein distance, if x and y ∈ R d are close to each other and if f : R d → R is Lipschitz continuous, then we can bound |f (x) -f (y)| by [f ] Lip |x -y|; whereas if f is simply measurable and bounded, then we cannot directly bound |f (x) -f (y)| in terms of |x -y|. Moreover the regularizing properties of the semi-group cannot be used in small time for the total variation distance.

Results in the literature focus on the Euler-Maruyama scheme. In [START_REF] Bally | The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function[END_REF] is proved the convergence for a fixed time horizon T > 0 and as N → ∞, where N is the number of steps in the Euler-Maruyama scheme on a finite horizon. More precisely, if σ 1 is elliptic and if b 1 and σ 1 are C ∞ with bounded derivatives (but b 1 and σ 1 are not supposed bounded themselves), then ([BT96, Theorem 3.1])

∀x ∈ R d , d TV (X x T , Xx,N T ) ≤ K(T )(1 + |x| Q ) N T q ,
where Xx,N T stands for the Euler scheme with N steps, where Q and q are positive exponents and where K is a non-decreasing function depending on b 1 and σ 1 . The common strategy of proof for such bounds is to use Malliavin calculus in order to perform an integration by parts and to use bounds on the derivatives of the density. However, we cannot infer a bound as T → 0 since we do not know whether K(T )/T q → 0 as T → 0 in general. In [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] are given bounds in small time and as N → ∞. Assuming that σ 1 is uniformly elliptic and that b 1 and σ 1 are bounded with bounded derivatives up to order 3, then ([GL08, Theorem 2.3])

∀t ∈ (0, T ], ∀x, y ∈ R d , |p(t, x, y) -pN (t, x, y)| ≤ K(T )T N t (d+1)/2 e -C|x-y| 2 /t ,
where p and pN denote the transition densities of X x and Xx,N respectively and where C is a positive constant depending on d and on the bounds on b 1 and σ 1 and on their derivatives. However, we cannot directly use this result for the total variation distance: taking N = 1 yields

d TV (X x t , Xx t ) = R d |p(t, x, y) -pN (t, x, y)|dy ≤ K(T )T t -1/2 R d 1 t d/2 e -C|x-y| 2 /t ,
giving a bound in t -1/2 which does not converge to 0 as t → 0. Moreover, [START_REF] Gobet | Sharp estimates for the convergence of the density of the Euler scheme in small time[END_REF] assumes that b 1 is bounded. [START_REF] Bencheikh | Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise[END_REF] focuses on the case where b 1 is bounded and measurable but not necessarily regular and where σ 1 is constant; it proves that the convergence in total variation of the Euler scheme on a finite horizon which is regularized with respect to the irregular drift b 1 and with step h, is of order √ h.

In the present paper, we first prove a convergence rate of order t 1/3 for d TV (X x t , Y x t ), provided that for i = 1, 2, σ i is elliptic, σ i and b i are Lipschitz-continuous with respect to their time variable and that σ i is C 2 b and b i is C 1 and Lipschitz-continuous with respect to their space variable. More generally, if we furthermore assume that σ is C 2r b , then we obtain a convergence rate of order t r/(2r+1) . Letting r → ∞, we also prove that if σ ∈ C ∞ b with some technical condition on the derivatives of the densities of the random variables X x t and Y x t , then the convergence rate is of order t 1/2 exp(C log(1/t)) which is "almost" t 1/2 . Moreover, we provide an example using the geometric Brownian motion where the convergence rate is exactly t 1/2 , thus showing that we cannot achieve better bounds in general. To prove the bound in t r/(2r+1) , we use a multi-step Richardson-Romberg extrapolation [RG11, Gil08, LP17], which is a method imported from numerical analysis that we use in our case for theoretical purposes. It relies on a Taylor expansion with null coefficients up to some high order. Such method can be used in more general settings with regularization arguments in order to improve the convergence rates (in our case, we improve t 1/3 into t r/(2r+1) ).

Interestingly, the difference between the drift coefficients b 1 -b 2 does not need to be small for our bounds to be valid. This is because the dominant term in d TV (X x t , Y x t ) comes from the the diffusion part.

Recent results (see [START_REF] Clement | Hellinger and total variation distance in approximating Levy driven SDEs[END_REF]) establish a convergence in small time at rate t 1/2 for the Euler scheme of certain classes of diffusions driven by stable Lévy processes, not directly including the Brownian case. This approach relies on Malliavin calculus techniques. In this work the "standard" drift is replaced in the Euler scheme by the flow of the associated (noiseless) ODE. This seems to be specific to Lévy driven SDEs. Adapting this approach to our general continuous framework is not as straightforward as could be expected and would deserve further investigations for future work.

The total variation distance is closely related to the estimation of the density of the solution to an SDE and this density satisfies a Fokker-Planck Partial Differential Equation PDE (8.3.2). If the drift is bounded, then the density and its partial derivatives admit sub-gaussian Aronson's bounds (see [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] and Section 8.3.1). However, giving estimates and bounds for the solution of the PDE in the case of unbounded drift appears to be more difficult, see [Lun97, Cer00, BL05]. Recent improvements have been made in [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF] using the parametrix method. Studying this case is useful to study the convergence in total variation of SDE's with unbounded drift, in particular for the Langevin equation, very popular in stochastic optimization, and which reads

dX t = -∇V (X t )dt + σ(X t )dW t ,
where in many cases, V : R d → R has quadratic growth and ∇V has linear growth (see for example Chapter 2).

In order to deal with unbounded b i , we propose two different methods. First, we use a localization argument and "cut" the drift b i into bi outside a compact set, so that we can use bounds from [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] for the bounded drift case. We use the Girsanov formula to explicit the dependence of these bounds in ∥ bi ∥ ∞ . A second method consists in using the density estimates from [MPZ21, Section 4] to improve the dependency with respect to the bounds in x. However this second approach relies on advanced parametrix methods which require further regularity assumptions on the coefficients of the SDE and which are not fully detailed for higher order derivatives. Our first method is clearly much more elementary, starting from a quite general bound established for any pair of integrable random vectors (see Theorem 8.2.7) and calling upon a standard regularization strategy which combined with a multistep procedure, seems to be at least quasi-optimal in a very general framework.

Notations

On top of the notations defined from page 1, we also use the following notations. For k ∈ N and if f :

R d → R is C k , we denote by ∇ k f : R d → (R d ) ⊗k its differential of order k. If f is Lipschitz continuous, we denote by [f ] Lip its Lipschitz constant. If f : (t, x) ∈ R × R d → f (t, x) is C k with respect to x, we still denote by ∇ k f its differential with respect to x.
If Z is a Markov process with values in R d , we denote, when it exists, its transition probability from x to y ∈ R d between times s < t, p Z (s, t, x, y).

Main results

We consider the two following SDEs in R d :

X x 0 = x ∈ R d , dX x t = b 1 (t, X x t )dt + σ 1 (t, X x t )dW t , t ∈ [0, T ], (8.2.1)
Y x 0 = x, dY x t = b 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t , t ∈ [0, T ], (8.2.2)
where T is a finite time horizon, b i :

R d → R d , σ i : R d → M d (R), i = 1, 2
, are Borel functions and W is a standard R d -valued Brownian motion defined on a probability space (Ω, A, P). The one-step Euler-Maruyama scheme of X, denoted X, is defined by Xx = Y x when Y x reads

dY x t = b 1 (0, x)dt + σ 1 (0, x)dW t , t ∈ [0, T ]. (8.2.3)
To allievate notations, we also define

∆b(x) := |b 1 (0, x) -b 2 (0, x)|, ∆σ(x) := |σ 1 (0, x) -σ 2 (0, x)|. ( 8.2.4)
Let us remark that if Y = X, then ∆b(x) = 0 and ∆σ(x) = 0. For g

: (t, x) ∈ [0, T ] × R d → g(t,
x) ∈ R q and r ∈ N, let us define the following assumptions:

• Lip t (g): g is Lipschitz continuous with respect to t, uniformly in x.

• g ∈ C r : g is differentiable with respect to x with continuous partial derivatives up to the order r.

• g ∈ C r b : g ∈ C r and is bounded with bounded partial derivatives up to the order r. • g ∈ C r b : g ∈ C r and has partial bounded derivatives up to the order r, but we do not assume that g is bounded itself.

• For σ :

[0, T ] × R d → M d (R), we say that σ is (uniformly) elliptic if ∃ σ0 > 0, ∀x ∈ R d , ∀t ∈ [0, T ], σ(t, x)σ(t, x) ⊤ ≥ σ2 0 I d . (8.2.5)
Theorem 8.2.1. Let X and Y be the solutions of the SDEs (8.2.1) and (8.2.2).

For i = 1, 2, assume Lip t (b i ), Lip t (σ i ), σ i ∈ C 2 b , b i ∈ C 1 b and σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2/3 + Ce c|x| 2 t 1/2 , (8.2.6)
where the positive constants C and c only depend on d, T , σ0 ,

∥σ i ∥ ∞ , [b i ] Lip , [σ i ] Lip and ∥∇ 2 σ i ∥ ∞ . In particular, if Y = X we have d TV (X x t , Y x t ) ≤ Ct 1/3 + Ce c|x| 2 t 1/2 . Theorem 8.2.2.
Let X and Y be the solutions of the SDEs (8.2.1) and (8.2.2).

For i = 1, 2, assume Lip t (b i ), Lip t (σ i ), that σ i ∈ C 2r b , b i ∈ C 1 b and that σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2r/(2r+1) + Ce c|x| 2 t 1/2 , (8.2.7)
where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants. In particular, if Y = X we have

d TV (X x t , Y x t ) ≤ Ct r/(2r+1) + Ce c|x| 2 t 1/2 .

Main results

Remark 8.2.3. In Theorems 8.2.1 and 8.2.2, we can actually improve the dependency of the constants in x in small time since we have more precisely:

∀t ∈ [0, Ce -2(2r+1)c|x| 2 ], d TV (X x t , Y x t ) ≤ C(t 1/2 + ∆σ(x)) 2r/(2r+1) .
Remark 8.2.4. We can adapt the framework of Theorem 8.2.2 to the study of SDEs with homogeneous vanishing noise, i.e. where for a > 0,

dX x t = b 1 (t, X x t )dt + aσ 1 (X x t ), dY x t = b 2 (t, Y x t ) + aσ 2 (Y x t )
and we identify the dependency in a as a → 0. Namely, with the same assumptions, for a > 0 small enough we have

d TV (X x t , Y x t ) ≤ Ce ca -1 |x| 2 t 1/2 + Ca -(d+r) (t 1/2 + ∆σ(x)) 2r/(2r+1) , (8.2.8)
where the constant C does not depend on a. This bound is obtained adapting the proof of Theorem 8.2.2 in Section 8.3.4 and using that if Z x is the martingale

dZ x t = aσ(Z x t )dW t then (Z x t ) ∼ ( Zx a 2 t )
where Zx t = σ( Zx t )dW t which does not depend on a. We can also improve the dependency in the initial condition x ∈ R d using [START_REF] Menozzi | Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift[END_REF], however at the expense of further regularity assumptions on b i and σ i , i = 1, 2. Theorem 8.2.5. Let X and Y be the solutions of the SDEs (8.2.1) and (8.2.2).

For i = 1, 2, assume Lip t (b i ), Lip t (σ i ), σ i ∈ C 2r+1 b , b i ∈ C 2r b and σ i is elliptic. Then ∀t ∈ [0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ C t 1/2 (1 + ∆b(x)) + ∆σ(x) + t(|b 1 | + |b 2 |)(0, x) 2r/(2r+1)
, (8.2.9) where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants. In particular, if Y = X, we have

d TV (X x t , Y x t ) ≤ Ct r/(2r+1) 1 + t 1/2 (|b 1 | + |b 2 |)(0, x)| 2r/(2r+1)
.

Remark 8.2.6. Choosing Y not to be the Euler-Maruyama scheme of X but a general SDE and expressing the bounds in the Theorems in terms of ∆b(x) and ∆σ(x) allows to extend our results to more general couples of diffusions with "close" coefficients, for example to SDE solvers other than the genuine Euler-Maruyama scheme. Also, it is helpful to study perturbed SDEs, for example if we consider

dX x t = b 1 (t, X x t )dt + a 1 (t)σ(X x t )dW t , dY x t = b 2 (t, Y x t )dt + a 2 (t)σ(Y x t )dW t (8.2.10)
where

|a 1 (t) -a 2 (t)| → 0 as t → ∞. Then we have d TV (X x t+s , Y x t+s ) ≤ Ce c|x| 2 s 1/2 + C(s 1/2 + |a 1 (t) -a 2 (t)|) 2r/(2r+1)
and we obtain different convergence rates as t → ∞ and s → 0, depending on (t, s). A noticeable example of this is the Langevin-simulated annealing SDE, see Chapter 2. Furthermore we remark that in Theorems 8.2.1 and 8.2.2, the bounds do not depend on ∆b, enhancing that the dominant term in the total variation comes from the diffusion part.

To improve the rate of convergence from t 1/3 in Theorem 8.2.1 to t r/(2r+1) in Theorem 8.2.2, we rely on a Richardson-Romberg extrapolation [START_REF] Fry | Ix. the approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam[END_REF][START_REF] Michael | Multilevel Monte Carlo path simulation[END_REF][START_REF] Lemaire | Multilevel Richardson-Romberg extrapolation[END_REF]; this argument can also be applied in a more general framework. The following proposition gives bounds on the total variation between two random vectors, knowing bounds on the L 1 -Wasserstein distance and bounds on the partial derivatives of the densities up to some order. Theorem 8.2.7. Let Z 1 and Z 2 be two random vectors in L 1 (R d ) and admitting densities p 1 and p 2 respectively with respect to the Lebesgue measure. Assume furthermore that p 1 and p 2 are C 2r with r ∈ N and that ∇ k p i ∈ L 1 (R d ) for i = 1, 2 and k = 1, . . . , 2r. Then we have

d TV (Z 1 , Z 2 ) ≤ C d,r W 1 (Z 1 , Z 2 ) 2r/(2r+1) R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ 1/(2r+1) (8.2.11)
where the constant C d,r depends only on d and on r.

If σ ∈ C ∞
b , then we also prove that we can "almost" get a convergence rate of order t 1/2 . Theorem 8.2.8. Let X and Y be the solutions of the SDEs (8.2.1) and (8.2.2).

For i = 1, 2, assume Lip t (b i ), Lip t (σ i ), that σ i ∈ C 2r b for every r ∈ N, b i ∈ C 1 b
, that σ i is elliptic and that ∆σ(x) = 0. Assume furthermore that if Z and V are the martingales dZ t = σ 1 (t, Z t )dW t and

dV t = σ 2 (t, Z t )dW t , then ∀r ∈ N, ∀t ∈ (0, T ], ∀x, y ∈ R d , ∥∇ 2r y p Z (0, t, x, y)∥ + ∥∇ 2r y p V (0, t, x, y)∥ ≤ C 2r t (d+2r)/2 e -c 2r |y-x| 2 /t with lim sup r→∞ C 2r c -d/2 2r 1/(2r) < ∞.
(8.2.12)

(see Theorem 8.3.1). Then ∀t ∈ (0, T ], ∀x ∈ R d , d TV (X x t , Y x t ) ≤ Ce c|x| 2 t 1/2 + Ct 1/2 e c √ log(1/t) , (8.2.13)
where the positive constants C and c only depend on d, T , σ0 , on ∥σ i ∥ ∞ , on the bounds on the derivatives of b i and σ i and on their Lipschitz constants.

Remark 8.2.9. Assumption (8.2.12) is satisfied in the case of a Brownian motion, which suggests that this assumption is satisfied in general provided that σ is "regular enough". Indeed, if dZ t = σdW t with σ ∈ M d (R) being non degenerate, then with Σ := σσ ⊤ we have for t > 0 and x, y ∈ R d :

p Z (0, t, x, y) = 1 det(Σ)t d/2 Φ Σ -1/2 y -x √ t , Φ(u) := 1 (2π) d/2 e -|u| 2 /2 .
Moreover for every r ∈ N and u ∈ R d we have

d r du r Φ(u) ≤ 1 (2π) d/2 | He r (|u|)|e -|u| 2 /2
where He r is the r th probabilist Hermite polynomial. Following [START_REF] Krasikov | New bounds on the Hermite polynomials[END_REF] we have

∀u ≥ 0, | He 2r (u)|e -u 2 /2 ≤ C2 -r √ r (2r)! 2 r! 2 ≤ C2 r √ r,
using the Stirling formula for the last inequality. On the other hand, using [AS64, 22.14.15] we have ∀u ≥ 0, | He 2r (u)|e -u 2 /4 ≤ 2 r+1 r!.

Then, for for every ε ∈ (0, 1],

| He 2r (u)|e -u 2 /2 = He 2r (u)e -u 2 /4 ε He 2r (u)e -u 2 /2 1-ε e -εu 2 /4 ≤ C (2 r r!) ε 2 r r 1/2 1-ε e -εu 2 /2 .
Then if we choose ε r = log -1 (r) for r ≥ 3, we have (r!) εr ≤ e εrr log(r) = e r so that d r du r Φ(u) ≤ C2 rεr e r 2 r r 1/2 e -εr|u| 2 /2 =: A r e -εr|u| 2 /2 and then for r ≥ 3 we have

∥∇ 2r y p Z (0, t, x, y)∥ ≤ ∥Σ -1/2 ∥ 2r det(Σ)t (d+2r)/2 d 2r du 2r Φ Σ -1/2 y -x √ t ≤ ∥Σ -1/2 ∥ 2r det(Σ)t (d+2r)/2 A r e -εr∥Σ -1 ∥|y-x| 2 /(2t)
where

∥Σ -1/2 ∥ 2r A r ε -d/2 r 1/(2r)
is bounded. Thus Assumption (8.2.12) is satisfied.

Remark 8.2.10. For the Euler-Maruyama scheme (8.2.3), with a slight abuse of notation, x is used both for the starting point and in the definition of the drift and diffusion coefficients. The transition density should be considered for constant drift and diffusion coefficients in this case. However the results remain valid as the Euler scheme is simply a Brownian process.

Proof of the Theorems

Recalls on density estimates for SDEs with bounded drift

We recall results on the bounds for the density of the solution of the SDE using the theory of partial differential equations. Let us consider a generic SDE:

Z x 0 = x ∈ R d , dZ x t = b Z (t, Z x t )dt + σ Z (t, Z x t )dW t , t ∈ [0, T ]. (8.3.1)
Then under regularity assumptions on b Z and on σ Z , the transition probability p Z exists and is solution of the backward Kolmogorov PDE:

p Z (t, t, x, •) = δ x , t ∈ [0, T ], ∂ s p Z (s, t, x, y) = ⟨b Z (s, x), ∇ x p Z (s, t, x, y)⟩ + 1 2 Tr σ ⊤ Z (s, x)∇ 2 x p Z (s, t, x, y)σ Z (s, x) (8.3.2)
for s < t ∈ [0, T ]. Moreover, p Z and its derivatives satisfy sub-gaussian Aronson's bounds: [START_REF] Friedman | Partial differential equations of parabolic type[END_REF], Chapter 9, Theorem 7). Let Z be the solution of (8.3.1) and let T > 0. Assume Lip t (b Z ) and Lip t (σ Z ), that b Z , σ Z ∈ C r b and that σ Z is elliptic. Then for every m 0 = 0, 1 and for every 0

Theorem 8.3.1 ([
≤ m 1 + m 2 ≤ r, ∇ m 0 +m 1 x ∇ m 2 y p Z exists and ∀s<t ∈ [0, T ], ∀x, y ∈ R d , ∥∇ m 0 +m 1 x ∇ m 2 y p Z (s, t, x, y)∥ ≤ C (t -s) (d+m 0 +m 1 +m 2 )/2 e -c|y-x| 2 /(t-s) , ( 8
(σ Z ), that b Z ∈ C r b , σ Z ∈ C r+1 b
and that σ Z is elliptic. Then for every 0 ≤ m ≤ r, ∇ m y p Z exists and

∀s < t ∈ [0, T ], ∀x, y ∈ R d , ∥∇ m y p Z (s, t, x, y)∥ ≤ C (t -s) (d+m)/2 e -c|y-x| 2 /(t-s) , (8.3.4)
where the constants C and c only depend on the bounds on b Z and σ Z and on their derivatives and on their Lipschitz constants, on the modulus of ellipticity of σ Z , on d and on T .

Preliminary results

In order to apply the bounds on the densities from Theorem 8.3.1 to Theorem 8.2.7, we first "cut" the drifts b 1 and b 2 on a compact set. That is, we instead consider the processes X and Y defined by

d X x t = bx 1 (t, X x t )dt + σ 1 (t, X x t )dW t , t ∈ [0, T ], (8.3.5) d Y x t = bx 2 (t, Y x t )dt + σ 2 (t, Y x t )dW t , t ∈ [0, T ], (8.3.6)
where bi , i = 1, 2 is defined as follows. We choose R > 0 and we consider a C ∞ decreasing function ψ : R + → R + such that ψ = 1 on [0, R 2 ] and ψ = 0 on [(R + 1) 2 , ∞) and we define bx i (t, y) := b i (t, y)ψ(|y -x| 2 ), so that bx i is bounded:

∀y ∈ R d , ∀t ∈ [0, T ], | bx i (t, y)| ≤ sup z∈B(x,R+1) |b i (t, z)| ≤ C(1 + |x|), (8.3.7) because b i is Lipschitz continuous. Lemma 8.3.3. Assume Lip t (b 1 ), Lip t (σ 1 ), b 1 ∈ C 1 b , σ 1 ∈ C 1 b . Then for every x ∈ R d and t ∈ [0, T ], d TV (X x t , X x t ) ≤ C(1 + |b 1 (0, x)| 2 )t. (8.3.8)
Proof. Let f : R d → R be measurable and bounded. We remark that on the event

{sup s∈[0,t] |X x s - x| 2 ≤ R 2 }, we have X x t = X x t , so that |Ef ( X x t ) -Ef (X x t )| ≤ 2∥f ∥ ∞ P sup s∈[0,t] |X x s -x| 2 > R 2 .
But using the inequality |u + v| 2 ≤ 2|u| 2 + 2|v| 2 we have

|X x t -x| 2 ≤ 2 t 0 b 1 (s, X x s )ds 2 + 2 t 0 σ 1 (s, X x s )dW s 2 ≤ 2t t 0 |b 1 (s, X x s )| 2 ds + 2 t 0 σ 1 (s, X x s )dW s 2 ≤ 4t[b 1 ] 2 Lip t 0 |X x s -x| 2 ds + 1 3 t 3 + 4t 2 |b 1 (0, x)| 2 + 2 t 0 σ 1 (s, X x s )dW s 2 so that E sup s∈[0,t] |X x s -x| 2 ≤ 4t[b 1 ] 2 Lip t 0 E sup u∈[0,s] |X x u -x| 2 ds + 4 3 [b 1 ] 2 Lip t 4 + 4t 2 |b 1 (0, x)| 2 + 2E sup s∈[0,t] s 0 σ 1 (u, X x u )dW u 2 .
Moreover using Doob's martingale inequality we have

E sup s∈[0,t] s 0 σ 1 (u, X x u )dW u 2 ≤ 4E t 0 σ 1 (u, X x u )dW u 2 = 4E t 0 σ 2 1 (u, X x u )du ≤ 4∥σ 1 ∥ 2 ∞ t.
Then we define the non-decreasing deterministic process S t := E sup s∈[0,t] |X x s -x| 2 and we get the differential inequality (using t ≤ T )

S t ≤ 4t T |b 1 (0, x)| 2 + 1 3 [b 1 ] 2 Lip T 3 + 2∥σ 1 ∥ 2 ∞ + 4t[b 1 ] 2 Lip t 0
S s ds, so the Gronwall lemma yields

S t ≤ 4t T |b 1 (0, x)| 2 + 1 3 [b 1 ] 2 Lip T 3 + 2∥σ 1 ∥ 2 ∞ e 2t 2 [b 1 ] 2 Lip ≤ C(1 + |b 1 (0, x)| 2 )t.
Using Markov's inequality, we have then

|Ef ( X x t ) -Ef (X x t )| ≤ 2∥f ∥ ∞ P sup s∈[0,t] |X x s -x| 2 > R 2 ≤ 2∥f ∥ ∞ C(1 + |b 1 (0, x)| 2 )t R 2 .
We can now apply Theorem 8.3.1 to X and to Y however the constants arising depend on the bound on ∥ bx i ∥ ∞ and thus on x. In order to deal with the dependency in ∥ bx i ∥ ∞ , we apply the Girsanov formula and reduce to the null drift case. Proposition 8.3.4. Let Z x be the solution of

Z x 0 = x, dZ x t = σ 1 (t, Z x t )dW t , t ∈ [0, T ]. (8.3.9) Assume Lip t (b 1 ), Lip t (σ 1 ), b 1 ∈ C 1 b , σ 1 ∈ C 1 b and σ 1 is elliptic. Then we have for every t ∈ [0, T ], x, y ∈ R d , p X (0, t, x, y) = p Z (0, t, x, y) + t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds, (8.3.10)
where X is defined in (8.3.5) and U x is defined as

U x s = exp s 0 ⟨g(u, Z x u ) bx 1 (u, Z x u ), dZ x u ⟩ - 1 2 s 0 ⟨g(u, Z x u ) bx 1 (u, Z x u ), bx 1 (u, Z x u )⟩du , (8.3.11) g = (σ 1 σ ⊤ 1 ) -1 . (8.3.12)
Proof. First, note that since σ 1 is elliptic and since bx 1 , σ 1 ∈ C 1 b , then p X and p Z exist as well as ∇ x p Z (Theorem 8.3.1). We then use [QZ04, Theorem 2.4] extended to non-homogeneous diffusion processes. Following [QZ04, Remark 2.5], since σ 1 is elliptic and bounded, the assumptions of [QZ04, Theorem 2.4] hold.

We also have the following bounds on the process U x . Lemma 8.3.5. With the same assumptions as in Proposition 8.3.4, for every p ≥ 2, x ∈ R d and t ∈ [0, T ] we have

E sup s∈[0,t] |U x s | p ≤ e Cp 2 ∥ bx 1 ∥ 2 ∞ t . (8.3.13)
Proof. We recall that for every q ≥ 1, the process (U x ) q is a martingale with

d(U x s ) q = q(U x s ) q g(s, Z x s ) bx 1 (s, Z x s ), σ 1 (s, Z x s )dW s . Thus, Doob's martingale inequality yields E sup s∈[0,t] |U x s | 2q ≤ Cq 2 ∥ bx 1 ∥ 2 ∞ E t 0 |U x s | 2q ds ≤ Cq 2 ∥ bx 1 ∥ 2 ∞ t 0 E sup u∈[0,s] |U x s | 2q ds.
So with U x 0 = 1 we obtain

E sup s∈[0,t] |U x s | 2q ≤ e Cq 2 ∥ bx 1 ∥ 2 ∞ t .
Lemma 8.3.6. With the same assumptions as in Proposition 8.3.4, we have for every x, y ∈ R d and t ∈ [0, T ],

t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ Ce C∥ bx 1 ∥ 2 ∞ e -c|y-x| 2 /t t (d-1)/2 . (8.3.14)
Proof. We use Theorem 8.3.1 on the process Z x , which yields bounds with constants depending on σ 1 but not on bx 1 . We obtain for every q ≥ 1 and for every s ∈ [0, t]:

E |∇ x p Z (s, t, Z x s , y)| q = R d |∇ x p Z (s, t, ξ, y)| q p Z (0, s, x, ξ)dξ ≤ C q (t -s) (q+(q-1)d)/2 R d 1 (s(t -s)) d/2 exp -c q |y -ξ| 2 t -s + |ξ -x| 2 s dξ ≤ C q t d/2 1 (t -s) (q+(q-1)d)/2 e -cq|y-x| 2 /t ,
where we used Lemma 8.5.1 in the appendix. Then for p -1 + q -1 = 1 and p ≥ 2, using the Hölder inequality we have

t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ ∥ bx 1 ∥ ∞ sup s∈[0,t] E|U x s | p 1/p t 0 (E |∇ x p Z (s, t, Z x s , y)| q ) 1/q ds ≤ ∥ bx 1 ∥ ∞ e Cp∥ bx 1 ∥ 2 ∞ t C q e -cq|y-x| 2 /t t d/(2q) t 0 ds (t -s) (1+(1-q -1 )d)/2 .
The integral in ds converges under the condition q < d/(d -1) if d > 1, and for any value of q > 1 if d = 1. Then performing the change of variable s = tu we obtain

t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds ≤ ∥ bx 1 ∥ ∞ e Cp∥ bx 1 ∥ 2 ∞ T C q e -cq|y-x| 2 /t t (d-1)/2 ≤ Ce C∥ bx 1 ∥ 2 ∞ e -c|y-x| 2 /t t (d-1)/2 .

Proof of Theorem 8.2.1

Lemma 8.3.7. We have for every x ∈ R d and t ∈ [0, T ]:

d TV (X x t , Y x t ) ≤ d TV (Z x t , V x t ) + Ce C|x| 2 t 1/2 ,
where

dZ x t = σ 1 (t, Z x t )dW t and dV x t = σ 2 (t, V x t )dW t . Proof. Let us write d TV (X x t , Y x t ) ≤ d TV (X x t , X x t ) + d TV ( X x t , Z x t ) + d TV (Z x t , V x t ) + d TV (V x t , Y x t ) + d TV ( Y x t , Y x t ),
where X and Y are defined in (8.3.5) and (8.3.6). Using Lemma 8.3.3 with (8.3.7), we have

d TV (X x t , X x t ) + d TV ( Y x t , Y x t ) ≤ C(1 + |x| 2 )t.
Using the formula (8.3.10) and the inequality (8.3.14), we have

d TV ( X x t , Z x t ) = R d |p X (0, t, x, y)-p Z (0, t, x, y)|dy = R d t 0 E U x s ⟨ bx 1 (s, Z x s ), ∇ x p Z (s, t, Z x s , y)⟩ ds dy ≤ Ce C∥ bx 1 ∥ 2 ∞ t 1/2 R d e -c|x-y| 2 /t t d/2 dy ≤ Ce C|x| 2 t 1/2 ,
where we used (8.3.7). The term d TV ( Y x t , V x t ) is treated likewise.

We now prove Theorem 8.2.1.

Proof. Let us introduce an artificial regularization. For ε > 0 and using Lemma 8.3.7 we have

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + d TV (Z x t , Z x t + √ εζ) + d TV (Z x t + √ εζ, V x t + √ εζ) (8.3.15) + d TV (V x t + √ εζ, V x t ) (8.3.16)
where ζ ∼ N (0, I d ) and is independent of the Brownian motion W .

• Let f : R d → R be measurable and bounded and let us define

φ : y ∈ R d → Ef (Z x t + y) = R d f (ξ + y)p Z (0, t, x, ξ)dξ = R d f (ξ)p Z (0, t, x, ξ -y)dξ. (8.3.17) Then φ is C 2 with ∇ 2 φ(y) = R d f (ξ)∇ 2 y p Z (0, t, x, ξ -y)dξ.
Moreover, using Theorem 8.3.1, we have

∥∇ 2 y p Z (0, t, x, ξ -y)∥ ≤ C t (d+2)/2 e -c|x-ξ+y| 2 /t ,
where the constants C and c do not depend on bx 1 . This implies that for every y ∈ R d ,

∥∇ 2 φ(y)∥ ∞ ≤ C∥f ∥ ∞ t -1 R d 1 t d/2 e -c|x-ξ+y| 2 /t dξ ≤ C∥f ∥ ∞ t -1 .
Then using the Taylor formula, for every y ∈ R d there exists ỹ ∈ (0, y) such that

φ(y) = φ(0) + ∇φ(0) • y + 1 2 ∇ 2 φ(ỹ) • y ⊗2
and then for some random ζ ∈ (0, ζ) we have

|Ef (Z x t + √ εζ) -Ef (Z x t )| = |Eφ( √ εζ) -φ(0)| = √ εE[∇φ(0) • ζ] + ε 2 E[∇ 2 φ( √ ε ζ) • ζ ⊗2 ] ≤ Cε∥f ∥ ∞ t -1 ,
where we used that E[∇φ(0

) • ζ] = ∇φ(0) • E[ζ] = 0. This way we obtain d TV (Z x t , Z x t + √ ϵζ) ≤ Cεt -1 . The term d TV (V x t , V x t + √ ϵζ) is treated likewise.
• Let f : R d → R be measurable and bounded and let us define

f ε : y → Ef (y + √ εζ) = 1 (2π) d/2 R d f (y + √ εξ)e -|ξ| 2 /2 dξ = 1 (2πε) d/2 R d f (ξ)e -|ξ-y| 2 /(2ε) dξ. (8.3.18) Then f ε is C 1 with ∇f ε (y) = 1 (2πε) d/2 R d f (ξ) ξ -y ε e -|ξ-y| 2 /(2ε) dξ = ε -1/2 (2π) d/2 R d f (y + √ εξ)ξe -|ξ| 2 /2 dξ = ε -1/2 E[f (y + √ εζ)ζ]
and then

[f ε ] Lip ≤ ∥f ∥ ∞ ε -1/2 E|ζ| ≤ C∥f ∥ ∞ ε -1/2 . (8.3.19) So that |Ef (Z x t + √ εζ) -Ef (V x t + √ εζ)| = |Ef ε (Z x t ) -Ef ε (V x t )| ≤ C∥f ∥ ∞ √ ε ∥Z x t -V x t ∥ 1 ≤ C∥f ∥ ∞ √ ε (t + t 1/2 ∆σ(x)), (8.3.20)
where we used Lemma 8.5.3 in the Appendix. This implies that

d TV (Z x t + √ εζ, V x t + √ εζ) ≤ Cε -1/2 (t + t 1/2 ∆σ(x)).
• Conclusion : Considering (8.3.15), we have

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + Cεt -1 + Cε -1/2 (t + t 1/2 ∆σ(x)).
We now choose ε = t(t + t 1/2 ∆σ(x)) 2/3 , so that

d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + C(t 1/2 + ∆σ(x)) 2/3 .

Proof of Theorem 8.2.2 using Theorem 8.2.7

We first prove Theorem 8.2.7.

Proof. Let f : R d → R be measurable and bounded, let ε > 0 and let ζ ∼ N (0, I d ) be independent of (Z 1 , Z 2 ). We have

|Ef (Z 1 ) -Ef (Z 2 )| ≤ Ef (Z 1 ) - r i=1 w i Ef ε/n i (Z 1 ) + r i=1 w i Ef ε/n i (Z 1 ) - r i=1 w i Ef ε/n i (Z 2 ) + r i=1 w i Ef ε/n i (Z 2 ) -Ef (Z 2 ) , (8.3.21)
where f ε is defined as in (8.3.18) and where the n i 's and the w i 's will be defined later.

Let φ be as defined in (8.3.17) replacing Z x t by Z 1 . Then, φ is differentiable up to the order 2r and for all k = 0, 1, . . . , 2r:

∇ k φ(y) = (-1) k R d f (ξ)∇ k p 1 (ξ -y)dξ.
Using the Taylor formula up to order 2r, for every y ∈ R d there exits ỹ ∈ (0, y) such that

φ(y) = φ(0) + 2r-1 k=1 ∇ k φ(0) k! • y ⊗k + ∇ 2r φ(ỹ) (2r)! • y ⊗2r .
Moreover, we have

∇ 2r φ(ỹ) • y ⊗2r ≤ C∥f ∥ ∞ |y| 2r R d ∥∇ 2r p 1 (ξ)∥dξ. (8.3.22)
Then there exists a random ζ ∈ (0, ζ) such that

Ef (Z 1 + √ εζ) -Ef (Z 1 ) = Eφ( √ εζ) -φ(0) (8.3.23) = 2r-1 k=1 ∇ k φ(0) k! ε k/2 • E[ζ ⊗k ] + E[∇ 2r φ( √ ε ζ) • ζ ⊗2r ] (2r)! ε r = r-1 k=1 ∇ 2k φ(0) (2k)! ε k • E[ζ ⊗2k ] + E[∇ 2r φ( √ ε ζ) • ζ ⊗2r ] (2r)! ε r =: r-1 k=1 β k (t)ε k + βr (t, ε)ε r , (8.3.24) because if k is odd, then E[ζ ⊗k ] = 0.
We now rely on a multi-step Richardson-Romberg extrapolation [LP17, Appendix A]. Let us denote the refiners n i = 2 i-1 and the auxiliary sequences and weights

u k := k-1 ℓ=1 (1 -2 -ℓ ) -1 , v k := (-1) k 2 -k(k+1)/2 u k+1 , w k := u k v r-k , k = 1, . . . , r. (8.3.25)
These weights are the unique solution to the r × r Vandermonde system r i=1

w i n -k i = 1 if k = 0, 0 else. , k = 0, 1, . . . ,
w i Ef (Z 1 + ε/n i ζ) -Ef (Z 1 ) = r i=1 w i r-1 k=1 β k (t)ε k n -k i + r i=1 w i βr (t, ε/n i )ε r n -r i = r-1 k=1 ε k β k (t) r i=1 w i n -k i + ε r r i=1 βr (t, ε/n i )w i n -r i = ε r r i=1 βr (t, ε/n i )w i n -r i , (8.3.27)
where we used (8.3.26) in the last equation. Now, using (8.3.22) we have

r i=1 βr (t, ε/n i )w i n -r i ≤ C∥f ∥ ∞ R d ∥∇ 2r p 1 (ξ)∥dξ r i=1 |w i |n -r i . Since u k → u ∞ = ℓ≥1 (1 -2 -ℓ ) -1 < ∞, the weights satisfy |w i | ≤ u 2 ∞ 2 -(r-i)(r-i+1)/2 , i = 1, . . . , r, so that r i=1 |w i | n r i ≤ u 2 ∞ r i=1 2 -(r-i)(r-i+1)/2 ≤ u 2 ∞ r i=1 2 (r-i)/2 = u 2 ∞ r-1 i=0 2 -i/2 ≤ C. (8.3.28)
As a consequence and since r i=1 w i = 1, we may write from (8.3.27)

Ef (Z 1 ) - r i=1 w i Ef ε/n i (Z 1 ) ≤ C∥f ∥ ∞ ε r R d ∥∇ 2r p 1 (ξ)∥dξ. (8.3.29)
The same way, we obtain

Ef (Z 2 ) - r i=1 w i Ef ε/n i (Z 2 ) ≤ C∥f ∥ ∞ ε r R d ∥∇ 2r p 2 (ξ)∥dξ.
On the other side, using (8.3.19) we have r i=1

w i Ef ε/n i (Z 1 ) - r i=1 w i Ef ε/n i (Z 2 ) ≤ C∥f ∥ ∞ √ ε W 1 (Z 1 , Z 2 ) r i=1 |w i |2 (i-1)/2 . (8.3.30)
Moreover, for every i = 1, . . . , r,

|w i |2 (i-1)/2 ≤ u 2 ∞ 2 -(r-i)(r-i+1
)/2+(i-1)/2 and then r i=1

|w i |2 (i-1)/2 ≤ u 2 ∞ r i=1 2 (i-1)/2 ≤ u 2 ∞ 2 r . (8.3.31)
Thus considering (8.3.21), we obtain for every ε > 0,

d TV (Z 1 , Z 2 ) ≤ Cε r R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ + Cε -1/2 W 1 (Z 1 , Z 2 ).
Optimizing in ε gives

ε ⋆ = W 1 (Z 1 , Z 2 )/(2r R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ) 2/(2r+1)
and then

d TV (Z 1 , Z 2 ) ≤ C d,r W 1 (Z 1 , Z 2 ) 2r/(2r+1) R d ∥∇ 2r p 1 (ξ)∥ + ∥∇ 2r p 2 (ξ)∥ dξ 1/(2r+1)
.

We

| βr (t, ε)| ≤ C 2r ∥f ∥ ∞ t -r E[|ζ| 2r ] (2r)! , C 2r := C 2r c -d/2 2r ,
where C 2r and c 2r are defined in (8.2.12) and where

E[|ζ| 2r ] = 2 r Γ(d/2 + r) Γ(d/2) = r-1 i=0 (d + 2i).
Using (8.3.28) we get

r i=1 βr (t, ε/n i )w i n -r i ≤ C C 2r ∥f ∥ ∞ t -r r-1 i=0 (d + 2i) (2r)!
and we obtain as in (8.3.29):

Ef (Z x t ) - r i=1 w i Ef ε/n i (Z x t ) ≤ 1 2 κ 1 ∥f ∥ ∞ ε r t -r , κ 1 := C C 2r r-1 i=0 (d + 2i) (2r)! Ef (V x t ) - r i=1 w i Ef ε/n i (V x t ) ≤ 1 2 κ 1 ∥f ∥ ∞ ε r t -r .
On the other hand, considering (8.3.30) and (8.3.31) with Lemma 8.5.3 with ∆σ(x) = 0 we have r i=1

w i Ef ε/n i (V x t ) - r i=1 w i Ef ε/n i (Z x t ) ≤ κ 2 ∥f ∥ ∞ √ ε t, κ 2 := C2 r . (8.3.33) We now minimize κ 1 ε r t -r + κ 2 ε -1/2 t in ε, giving ε ⋆ = t (2r+2)/(2r+1) (2rκ 1 ) 2/(2r+1) κ 2/(2r+1) 2
and then

κ 1 ε r ⋆ t -r + κ 2 ε -1/2 ⋆ t ≤ Cκ 2r/(2r+1) 2 κ 1/(2r+1) 1 t r/(2r+1)
with as r → ∞:

κ 2r/(2r+1) 2 κ 1/(2r+1) 1 ∼ C 1/(2r+1) 2r r-1 i=0 (d + 2i) 1/(2r+1) 1 (2r)! 1/(2r+1) 2 2r 2 /(2r+1)
(d + 2i) 1 (2r+1) = exp r 2r+1 1 r r-1 i=0 log(d + 2i) ≤ exp r 2r+1 log(d+(r-1)) ≤ √ d+r-1, 1 (2r)! 1/(2r+1) ∼ e 2r , lim sup r→∞ C 1/(2r+1) 2r < ∞
where we used Assumption (8.2.12), so that

κ 2r/(2r+1) 2 κ 1/(2r+1) 1 ≤ C √ d + r -1 e 2r 2 r .
Then we have d TV (Z x t , V x t ) ≤ C2 r r -1/2 t r/(2r+1) and we choose r(t) = ⌊log 1/2 (1/t)⌋ so that as t → 0, d TV (Z x t , V x t ) ≤ Ct 1/2 exp C log(1/t) .

Counterexample

In this section we give a counter-example showing that we cannot achieve a bound better than t 1/2 in general. More specifically, we show that we cannot achieve a bound better than t 1/2 for the total variation between an SDE and its Euler-Maruyama-scheme in general. For x > 0 and σ > 0, let us consider the one-dimensional process

Y x t =
(Y x t , Ȳ x t ) = 1 √ 2πσ 2 t R exp - log 2 (y/x) 2σ 2 t y -1 1 y≥0 -exp - (y -x -xtσ 2 /2) 2 2σ 2 x 2 t x -1 dy ≥ 1 √ 2πσ 2 ∞ -x/ √ t 1 x + √ ty exp - log 2 (1 + √ ty/x) 2σ 2 t - 1 x exp - (y -x √ tσ 2 /2) 2 2σ 2 x 2
dy.

But we have as (t, y) → 0:

1 1 + √ ty/x exp - log 2 (1 + √ ty/x) 2σ 2 t -exp - (y -x √ tσ 2 /2) 2 2σ 2 x 2 = (1 - √ ty/x + O(ty 2 )) exp - 1 2σ 2 t ( ty 2 x 2 - t 3/2 y 3 x 3 + O(t 2 y 4 )) -exp - y 2 2σ 2 x 2 - tσ 2 8 + √ ty 2σ 2 x = e -y 2 2σ 2 x 2 (1 - √ ty/x + O(ty 2 )) 1 + √ ty 3 2σ 2 x 3 + O(ty 4 ) -1 + √ ty 2σ 2 x - tσ 2 8 + O(t 2 ) + O(ty 2 ) = e -y 2 2σ 2 x 2 - √ ty x - √ ty 2σ 2 x + √ ty 3 2σ 2 x 3 + tσ 2 8 + O(ty 2 ) + O(t 2 ) .
Thus there exists ϵ > 0 and t 0 such that for every t ≤ t 0 :

d TV (Y x t , Ȳ x t ) ≥ 1 √ 2πσ 2 x 2 e -ε 2 2σ 2 x 2 √ t 2 ε -ε - y x - y 2σ 2 x + y 3 2σ 2 x 3 dy, so that d TV (Y x t , Ȳ x t
) is of order t 1/2 as t → 0.

However, the process Y does not satisfy the assumptions of Theorem 8.2.2 as its noise coefficient is not elliptic neither bounded on (0, ∞). We then prove the following result. Proposition 8.4.2. There exists a diffusion process X on R with C 1 and Lipschitz continuous drift, with C ∞ b and elliptic diffusion coefficient σ and there exists T > 0 and ε ∈ (0, 1) such that

∀t ∈ [0, T ], ∀x ∈ (ε, ε -1 ), d TV (X x t , Xx t ) ≥ C x t 1/2
where X is the Euler-Maruyama scheme of X and where the positive constant C x depends on x.

Proof. We construct from the geometric Brownian motion Y defined in (8.4.1), a process X with elliptic and bounded drift and such that d TV (X x t , Xx t ) ≥ C x t 1/2 . For ε ∈ (0, 1/2), let us consider ψ : R → R + a C ∞ b approximation of

ψ : x ∈ R -→      x if x ∈ [ε, ε -1 ], ε if x ≤ ε ε -1 if x ∈ [ε -1 , ∞) such that ψ = ψ on [2ε, ε -1 /2] ∪ (-∞, ε/2] ∪ [2ε -1 , ∞).
Then we define the process with elliptic and bounded noise coefficient

dX x t = - σ 2 2 X x t dt + σψ(X x t )dW t .
Then for x ∈ (2ε, ε -1 /2) we have Xx t = Ȳ x t and

P(Y x t ̸ = X x t ) ≤ P sup s∈[0,t] Y x s ≥ ε -1 /2 + P inf s∈[0,t] Y x s ≤ 2ε .
With a proof similar to the proof of Lemma 8.3.3, we show that

P sup s∈[0,t] Y x s ≥ ε -1 /2 ≤ C x,ε t.
Moreover, we remark that (Y x ) -1 ∼ x -2 Y x in law so

P inf s∈[0,t] Y x s ≤ 2ε = P sup s∈[0,t] (Y x s ) -1 ≥ ε -1 /2 = P sup s∈[0,t] Y x s ≥ x 2 ε -1 /2 ≤ C x,ε t.
Then we obtain

d TV (X x t , Xx t ) ≥ d TV (Y x t , Ȳ x t ) -d TV (X x t , Y x t ) ≥ C x √ t.
Remark 8.4.3. We could also consider the process X with "cut" bounded drift b and get the same bounds, proving then that we cannot achieve better bounds in general than the ones established in Theorem 8.2.8 even if we assume that b is bounded.

Appendix

Lemma 8.5.1 [START_REF] Friedman | Partial differential equations of parabolic type[END_REF], Chapter 9, Lemma 7). For a > 0, 0 < u < t ≤ T , x ∈ R d , ξ ∈ R d , let

I a := R d 1 (u(t -u)) d/2 exp -a |x -y| 2 t -u + |y -ξ| 2 u dy.
Then there exists a constant C > 0 depending only on d and T such that for every 0 < ε < 1,

I a ≤ C (εat) d/2 exp -a(1 -ε) |x -ξ| 2 t .
Let us recall [PP23, Lemma 3.4(a)], with an immediate adaptation to the non-homogeneous case.

Lemma 8.5.2. Let Z be solution to the generic SDE:

Z x 0 = x ∈ R d , dZ x t = b(t, Z x t )dt + σ(t, Z x t )dW t , t ∈ [0, T ],
where b and σ are Lipschitz continuous in (t, x) and where σ is bounded. Then for p ≥ 1, Proof. We first deal with the case p ≥ 2. We have

∀t ∈ [0, T ], ∀x ∈ R d , ∥Z x t -x∥ p ≤ C(p, T, [b] Lip , [σ] Lip , ∥σ∥ ∞ ) t|b(0, x)| + t 1/
∥X x t -Y x t ∥ p ≤ t 0 (b 1 (s, X x s ) -b 2 (s, Y x s ))ds p + t 0 (σ 1 (s, X x s ) -σ 2 (s, Y x s ))dW s p ≤ t 0 (b 1 (s, X x s ) -b 1 (0, x))ds p + t∆b(x) + t 0 (b 2 (s, Y x s ) -b 2 (0, x))ds p + t 0 (σ 1 (s, X x s ) -σ 1 (0, x))dW s p + t 0 ∆σ(x)dW s p + t 0 (σ 2 (s, Y x s ) -σ 2 (0, x))dW s p
But using the Burkholder-Davis-Gundy and the generalized Minkowski inequalities, we have

t 0 (σ 1 (s, X x s ) -σ 1 (0, x))dW s p ≤ C BDG p [σ 1 ] Lip t 0 |(s, X x s ) -(0, x)| 2 ds 1/2 p/2 ≤ C BDG p [σ 1 ] Lip t 0 ∥(s, X x s ) -(0, x)∥ 2 p ds 1/2 ≤ C(t + t 3/2 |b 1 (0, x)|),
where C BDG p is a constant which only depends on p and where we used Lemma 8.5.2. So that

∥X x t -Y x t ∥ p ≤ [b 1 ] Lip t 0 ∥(s, X x s ) -(0, x)∥ p ds + [b 2 ] Lip t 0 ∥(s, Y x s ) -(0, x)∥ p + t∆b(x) + C(t + t 3/2 (|b 1 | + |b 2 |)(0, x)) + ∆σ(x) √ t∥W 1 ∥ p ≤ C t(∆b(x) + 1) + t 3/2 (|b 1 | + |b 2 |)(0, x) + ∆σ(x) √ t
which completes the proof for p ≥ 2. For p ∈ [1, 2), the inequality is still true remarking that ∥ • ∥ p ≤ ∥ • ∥ 2 .

Introduction

Stochastic Volterra equations (SVE) have recently attracted much attention in the mathematical finance community in the context of rough volatility modelling, which is more able to reproduce some features of asset prices [ALV07, GJR18, EEFR18, JR20, Fuk17, Fuk21]. SVEs have also been introduced with regular (non-singular) kernel for modelling in population dynamics, biology and physics [START_REF] Salah-Eldin | Stochastic differential systems with memory: Theory, examples and applications[END_REF], in order to generalize modelling to non-Markovian stochastic systems with some memory effect. They where also motivated in particular by the physics of heat transfer [START_REF] Gripenberg | Encyclopedia of Mathematics and its Applications[END_REF] and have been mathematically studied since [START_REF] Berger | Volterra equations with Itô integrals[END_REF][START_REF] Protter | Volterra equations driven by semimartingales[END_REF]. Applications, e.g. pricing options in financial practice, require numerical methods to simulate the solution of the SVE, such as simulation through Euler-Maruyama schemes.

In the present paper, we give bounds for the weak error of the Euler-Maruyama scheme with N steps of an SVE on a finite time interval [0, T ] in the case where the kernel is non-singular. We consider two different Euler schemes: one where the kernel is not discretized, thus requiring the simulation of a large Gaussian matrix with covariance, and another one where the kernel is discretized, thus requiring only the simulation of (independent) Brownian increments.

A first bound on the weak error can be obtained from bounds on the strong error, however this is sub-optimal in general. For example, for Stochastic Differential Equations (SDE) the strong error is of order O(1/ √ N ) but the weak error is of order O(1/N ). Such bounds get even worse in the case of SVE with fractional kernel, giving a weak error bounded by the strong error which is order O(N -H ), where H ∈ (0, 1/2) is the Hurst parameter of the fractional kernel and is small (H ≃ 0.1) in many financial applications. In [START_REF] Richard | Discrete-time simulation of stochastic Volterra equations[END_REF] are given bounds for the weak error for the multi-level Euler-Maruyama scheme, however the authors only assume that the weak error is bounded by the strong error (see [RTY21, Section 2.3]). In [START_REF] Fukasawa | Limit distributions for the discretization error of stochastic Volterra equations[END_REF] is shown that the strong error is exactly of order H and the authors give the expression of the limit law of the (rescaled) strong error. In [START_REF] Gassiat | Weak error rates of numerical schemes for rough volatility[END_REF] are given weak error rates from some rough volatility models and are proved to be of order (3H + 1/2) ∧ 1, yielding significantly better bounds in the case where H is close to 0. However the results are valid only for some special cases (semilinear or cubic test function), hinting that obtaining general results for fractional processes is difficult. More recently, [START_REF] Peter | Weak error estimates for rough volatility models[END_REF] also proved that the weak error rate is of order (3H + 1/2) ∧ 1 for some class of stochastic rough volatility models, including the rough Bergomi and the rough Stein-Stein models. We recall that a rough (resp. Volterra) stochastic volatility model is a special case of a singular (resp. non-singular) two-dimensional SVE, where the first process is an asset price satisfying dS t = S t √ V t dB t for some Brownian motion B and where the second process (V t ) is the volatility satisfying some rough (resp. Volterra) stochastic equation, then giving for the joint process a matrix kernel K being diagonal and constant on its first coordinate.

Our main result states that for SVEs with non-singular kernel, the weak error of the Euler-Maruyama scheme is of order O(1/N ), which is the same rate as in the classic SDE case, under regularity assumptions on the coefficients and the kernel: we assume that the kernel is defined on the whole interval [0, T ] and is C 2 , that the drift and the diffusion coefficients are C 4 , bounded with bounded derivatives. Our strategy of proof consists in adapting the domino method from [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF] to the SVE case. In the SDE case and for a function f : R d → R, the domino strategy consists in a step-by-step decomposition of the weak error to produce an upper bound as follows:

|Ef ( Xx T ) -Ef (X x T )| = | Ph • • • • • Ph f (x) -P t N f (x)| ≤ N k=1 Ph • • • • • Ph • ( Ph -P h ) • P t N -t k f (x) , (9.1.1)
where h = T /N , t k = kh, X is the solution to the SDE and X is the corresponding Euler-Maruyama scheme, P and P are the semi-group operators associated to X and X respectively.

Then showing that with g := P tn-t k f , the short term weak error ( Ph -P h )g(x) is of order O(1/N 2 ), the sum in (9.1.1) is then of order O(1/N ). An elementary proof in the SDE case using the domino method can be found in [START_REF] Pagès | Numerical probability. Universitext[END_REF]Section 7.6]. This strategy fundamentally relies on a Markov semi-group and thus cannot be directly applied to the SVE case, as the 9.2. Setting and main results future trajectory of X depends on the whole previous trajectory in this last case. Instead, we consider the solution of an SVE as a Markov process on the infinite dimensional state space of trajectories Ω and then we define a infinite-dimensional Markov semi-group, allowing us to use the previously introduced domino method.

We then show that the weak error in small time is of order O(1/N 2 ) by establishing a Itō type formula for functionals g : Ω → R, involving the Fréchet derivatives of g. Such approach involving the derivatives of functionals on infinite dimensional state space and establishing Itō formula for SVEs was developed in [START_REF] Dupire | Functional Itô calculus[END_REF] and [START_REF] Viens | A martingale approach for fractional Brownian motions and related path dependent PDEs[END_REF], however in our case we only consider the infinite dimensional state space of continuous paths instead of càdlàg paths, since we consider a different semi-group.

Using a hybrid approach, combining ideas from both finite and infinite dimensional settings, the Itō formula with a finite dimensional Brownian motion on the one side, and the Fréchet derivatives of path functionals and the Markov property on an infinite dimensional state space on the other side, we establish the weak convergence rate of the Euler-Maruyama for SVEs. Studying the non-singular case by adapting the classic domino method to path-dependent setting constitutes a first step for studying weak error rates for fractional SVEs as we believe our method can be adapted to the singular case. We shall conduct such study in a next paper.

We give numerical evidence of the convergence rate we obtained on a Monte Carlo option pricing problem with a stochastic volatility model where the volatility follows some non-singular SVE. Proving weak error rates allows to design weighted and unweighted multi-level Richardson-Romberg extrapolation estimators [Gil08, GS14, LP17] that exploit the faster convergence of the weak error in comparison with the strong error; such application is particularly critical for the Monte Carlo simulation of Volterra and rough stochastic equations where the Vanilla Euler-Maruyama scheme has large time complexity (N 2 ).

The article is organized as follows. In Section 9.2 we give the precise setting and assumptions of the problem we consider, in particular the regularity assumptions on the coefficients and on the kernel of the SVE, and we state our main theorem. In Section 9.3, we give general results on random paths (φ t u ) u≥0,t∈[0,T ] which are adapted with respect to t, in particular we establish an Itō formula for infinite dimensional functionals g : Ω → R. The proof of the theorem is given in Section 9.4. Considering (9.1.1), the proof of our main result is decomposed into three parts: we prove that the short term error is of order O(1/N 2 ), applying Itō formula for a regular functional g : Ω → R as in the classic proof in the SDE case. Secondly we show that with g being the concatenation of discrete kernels applied to f , then the functional g is indeed differentiable in the Fréchet meaning and with bounded Fréchet derivatives. Lastly, in Section 9.5, we empirically check the weak convergence rate for some SVE model with non-singular kernel.

Setting and main results

Setting

Let us consider the following SVE in R d , d ∈ N:

X t = X 0 + t 0 K 1 (t, s)b(X s )ds + t 0 K 2 (t, s)σ(X s )dW s , t ∈ [0, T ], (9.2.1)
where (W t ) is a standard Brownian motion in R q 3 defined on some probability space (Ω, A, P), where b :

R d → R q 1 , K 1 : [0, T ] 2 → M d,q 1 (R), σ : R d → M q 2 ,q 3 (R), K 2 : [0, T ] 2 → M d,q 2 (R),
and where q 1 , q 2 , q 3 ∈ N and for a, b ∈ N, M a,b (R) denotes the set of a × b matrices with coefficients in R. We denote by (F t ) t∈[0,T ] the filtration generated by the Brownian motion. The starting point X 0 ∈ R d is fixed. Let us make the following assumptions on the coefficients b and σ and on the kernels K 1 and K 2 .

For (A, d A ) and (B, d B ) two metric spaces and k ∈ N, we consider the following sets of functions from A to B: When there is no ambiguity on the spaces, we also use the notations

• C k (A,
C k , C k b and C k b respectively. Assumption 9.2.1. (i) K 1 ∈ C 2 ([0, T ] 2 , M d,q 1 (R)) and K 2 ∈ C 4 ([0, T ] 2 , M d,q 2 (R)), which
guarantees that K 1 (resp. K 2 ) is bounded with bounded derivatives up to order 2 (resp. 4).

(ii) b ∈ C 5 b (R d , R q 1 ) and σ ∈ C 5 b (R d , M q 3 ,q 2 (R)). Then Assumption 9.2.1 guarantees that the solution of (9.2.1) is well defined (see for example [AJ18, Lemma 5.29]).

To simplify the notations and for more readability of the proofs, we assume hereafter that all the objects considered are one-dimensional, i.e. that d = q 1 = q 2 = q 3 = 1. However the main results in Section 9.2.2 remain valid for any (finite) dimensions d, q 1 , q 2 and q 3 , re-writing the proofs by replacing the the one-dimensional products by matrix products and writing them as sums over indices.

Let us define the Euler-Maruyama scheme associated to (9.2.1). For N ∈ N, we define the time step and the regular subdivision h := T /N, t k := kT /N, k ∈ {0, . . . , N } (9.2.2)

and Xt = X 0 + t 0 K 1 (t, s)b( Xs )ds + t 0 K 2 (t, s)σ( Xs )dW s , t ∈ [0, T ], (9.2.3)
where for s ∈ [0, T ], we define s = ⌊s/h⌋h. The solution of (9.2.3) can be recursively simulated as

Xt = X 0 + k j=0 t j+1 ∧t t j K 1 (t, s)b( Xt j )ds + k j=0 t j+1 ∧t t j K 2 (t, s)σ( Xt j )dW s , t ∈ [t k , t k+1 ],
where the integrals ( t j+1 t j K 2 (t, s)dW s ) j can be simulated on the discrete grid (t k ) 0≤k≤N by generating the independent sequence of Gaussian vectors

t j+1 t j K 2 (t k , s)dW s k=j,...,N
, j = 0, . . . , N -1, using the Cholesky decomposition of the covariance matrix

t j+1 t j K 2 (t k 1 , s)K 2 (t k 2 , s)ds k 1 ,k 2 =j,...,N .
We also define the Euler-Maruyama scheme associated to (9.2.1) with discretization of the kernels as We extend the definition of K 1 and K 2 on R + × R + such that for i = 1, 2, K i (t, s) = 0 for (t, s) / ∈ [0, 2T ] × [0, 2T ] and such that K i is still bounded with bounded derivatives up to order 2.

⇀ X t = X 0 + t 0 K 1 (t, s)b( ⇀ X s)ds + t 0 K 2 (t, s)σ( ⇀ X s)dW s , t ∈ [0, T ]. ( 9
In this paper, we use the notation C to denote a positive real constant, which may change from line to line. The constant C depends on the parameters of the problem: the coefficients and the kernels of the SVE, the time horizon T .

Main results

Theorem 9.2.2. Let f : R d → R be C 5 b and assume Assumption 9.2.1. Then we have

E[f ( XT )] -E[f (X T )] = O (1/N ) , (9.2.5) E[f ( ⇀ X T )] -E[f (X T )] = O (1/N ) .
(9.2.6) Remark 9.2.3.

• Since X, X and ⇀ X satisfy (9.2.1), (9.2.3) and (9.2.4) respectively with K 1 , K 2 , b and σ being bounded, we obtain

E|X T | + E| XT | + E| ⇀ X T | < +∞.
Since f is Lipschitz-continuous, we get

E|f (X T )| + E|f ( XT )| + E|f ( ⇀ X T )| < +∞.
• The strategy of proof we develop in Section 9.4 does not allow us to give weak error bounds for path-dependent functionals.

• We prove that the weak order of convergence of the Euler-Maruyama scheme for SVEs with regular kernels is the same as for SDEs, however the computation time for this scheme for SVEs is of order N 2 , against order N for SDEs.

Preliminary results on infinite dimensional paths

State space and path derivatives

For T ′ ∈ R + , we consider the infinite dimensional state space Ω T ′ being the space of R-valued continuous trajectories on [0, ∞) with support included in [0, T ′ ], with the topology of the supremum norm. For ω ∈ Ω T ′ such that ω is C 1 , we denote ω its derivative. If ω is Lipschitzcontinuous, we denote [ω] Lip its Lipschitz constant.

For g : Ω T ′ → R and for ω ∈ Ω T ′ , we define, when it exists, ∇g(ω) as the Fréchet derivative of g with respect to ω, which is a linear operator on Ω T ′ :

g(ω + η) = g(ω) + ⟨∇g(ω), η⟩ + o(∥η∥ ∞ ), η ∈ Ω T ′ .
More generally, for ℓ ∈ N we define, when it exists, the derivative of g of order ℓ recursively as the ℓ-multilinear operator on Ω ⊗ℓ T ′ :

⟨∇ ℓ-1 g(ω+η 1 ), ℓ j=2 η j ⟩ = ⟨∇ ℓ-1 g(ω), ℓ j=2 η j ⟩+⟨∇ ℓ g(ω), ℓ j=1 η j ⟩+o(∥η 1 ∥ ∞ ), η i ∈ Ω T ′ , i = 1, . . . , ℓ.
We use the notation ⊗ only to enhance the multilinearity of ∇ ℓ g.

Remark 9.3.1. The path derivative can be made explicit in some simple cases:

• If g(ω) = g(ω u 0 ) for some fixed u 0 ∈ R + and for some g : R → R, then we have

⟨∇ ℓ g(ω), ℓ j=1 η j ⟩ = ∇ ℓ g(ω u 0 ) • ℓ j=1 η j u 0 . (9.3.1) • If g(ω) = T ′ 0 g(ω u
)du for some g : R → R, then we have Then there exists a continuous modification φ of φ such that

⟨∇ ℓ g(ω), ℓ j=1 η j ⟩ = T ′ 0 ∇ ℓ g(ω u ) • ℓ j=1 η j u du. ( 9
E sup u≥0 φ u 2 ≤ C(T ′ ) 2 ∥∂ 1 F ∥ 2 ∞ E⟨M ⟩ r . (9.3.3) Proof. For u 1 , u 2 ∈ [0, T ′ ] we have E|φ u 1 -φ u 2 | 2 = E r 0 (F (u 1 , s) -F (u 2 , s))dM s 2 = E r 0 |F (u 1 , s) -F (u 2 , s)| 2 d⟨M ⟩ s ≤ |u 1 -u 2 | 2 ∥∂ 1 F ∥ 2 ∞ E⟨M ⟩ r ,
so that using the Kolmogorov continuity theorem (9.6.1), there exists a modification φ of φ which is almost surely α-Hölder for every α ∈ (0, 1/2), and taking for example α = 1/4 we have

E sup u 1 ,u 2 ∈[0,T ′ ],u 1 ̸ =u 2 | φ u 1 -φ u 2 | |u 1 -u 2 | 1/4 2 ≤ C(T ′ ) 3/2 ∥∂ 1 F ∥ 2 ∞ E⟨M ⟩ r ,
where C is an universal constant, so that taking u 1 = u and u 2 = 0 with φ 0 = 0 we obtain

E sup u≥0 φ u 2 = E sup u∈[0,T ′ ] φ u 2 ≤ C(T ′ ) 2 ∥∂ 1 F ∥ 2 ∞ E⟨M ⟩ r . ∀η 1 , η 2 ∈ Ω, ⟨∇ 2 G(ω), η 1 ⊗ η 2 ⟩ = 1≤i,j≤n ∂ ij G(ω u 1 , . . . , ω un )η 1 u i η 2 u j ,
and then (9.3.6) directly comes from the classic Itō formula.

Then, let us define for n ∈ N the regular subdivision of [0, T ′ ]: u n i = iT ′ /n, 0 ≤ i ≤ n and for ω ∈ Ω T ′ we define ω n as the affine interpolation of ω on the subdivision (u n i ) i i.e. ω n is equal to the affine interpolation on [0, T ′ ] and then ω n and ω are both equal to 0 on [T ′ , ∞); we also define G n as for every ω ∈ Ω T ′ , G n (ω) = G(ω n ). Then for every ω ∈ Ω 2T we have G n (ω) = G n (ω u 1 , . . . , ω un ) where G n : R n → R is the composition of the affine interpolation L n : R n → Ω 2T , which is a bounded linear operator, and of G, so is C 2 and then (9.3.6) is true for G n . Moreover for every ω, η ∈ Ω T ′ such that ω is Lipschitz-continuous we have

|G(ω) -G n (ω)| = |G(ω) -G(ω n )| ≤ ∥∇G∥ ∞ ∥ω -ω n ∥ ∞ ≤ C∥∇G∥ ∞ [ω] Lip /n.
Moreover, remarking that the affine interpolation is a bounded linear operator, we get that G n is also differentiable with

⟨∇G n (ω), η⟩ = ⟨∇G(ω n ), η n ⟩ so that ⟨∇G(ω), η⟩ -⟨∇G n (ω), η⟩ ≤ ∥∇ 2 G∥ ∞ ∥ω -ω n ∥ ∞ ∥η∥ ∞ + ∥∇G∥ ∞ ∥ω∥ ∞ ∥η -η n ∥ ∞ . (9.3.7) Moreover for η 1 , η 2 ∈ Ω T ′ we have ⟨∇ 2 G n (ω), η 1 ⊗ η 2 ⟩ = ⟨∇ 2 G(ω n ), (η 1 ) n ⊗ (η 2 ) n ⟩ so that ⟨∇ 2 G(ω), η 1 ⊗ η 2 ⟩ -⟨∇ 2 G n (ω), η 1 ⊗ η 2 ⟩ ≤ ∥∇ 3 G∥ ∞ ∥ω -ω n ∥ ∞ ∥η 1 ∥ ∞ ∥η 2 ∥ ∞ + ∥∇ 2 G∥ ∞ ∥ω∥ ∞ ∥η 1 -(η 1 ) n ∥ ∞ ∥η 2 ∥ ∞ + ∥ω∥ ∞ ∥η 1 ∥ ∞ ∥η 2 -(η 2 ) n ∥ ∞ (9.3.8) Writing (9.3.6) with G n gives G n (φ t ) = G n (φ 0 ) + t 0 ⟨∇G((φ s ) n ), Z n 1 (•, s)⟩ds + t 0 ⟨∇G((φ s ) n ), Z n 2 (•, s)⟩dW s + 1 2 t 0 ⟨∇ 2 G((φ s ) n ), Z n 2 (•, s) ⊗2 ⟩ds, (9.3.9)
and we have

E|G n (φ t ) -G(φ t )| 2 ≤ 2∥G∥ ∞ E|G n (φ t ) -G(φ t )| 2 ≤ C∥G∥ ∞ ∥∇G∥ ∞ E[φ t ] Lip n -1 with φ t being C 1 with φt u = φt 0 + t 0 ∂ 1 Z 1 (u, s)ds + t 0 ∂ 1 Z 2 (u, s)dW s ,
where the interchange is ensured by the stochastic Fubini theorem. But following Lemma 9.3.2 with assumption (9.3.5), E[φ t ] Lip < ∞, so that G n (φ t ) converges to G(φ t ) in L 2 . We proceed the same way for G n (φ 0 ). Moreover, we have

E t 0 ⟨∇G((φ s ) n ), Z n 1 (•, s)⟩ds - t 0 ⟨∇G(φ s ), Z 1 (•, s)⟩ds ≤ t 0 E|⟨∇G(φ s ), (Z n 1 (•, s) -Z 1 (•, s))⟩|ds + t 0 E|⟨∇G((φ s ) n ) -∇G(φ s ), Z n 1 (•, s))⟩|ds ≤ C∥∇G∥ ∞ E∥∂ 1 Z 1 ∥ ∞ n -1 + C∥∇ 2 G∥ ∞ t 0 E [φ s ] Lip ∥Z 1 ∥ ∞ ds n -1 ≤ C∥∇G∥ ∞ E∥∂ 1 Z 1 ∥ ∞ n -1 + C∥∇ 2 G∥ ∞ t 0 E[φ s ] 2 Lip E∥Z 1 ∥ 2 ∞ 1/2 ds n -1 -→ n→∞ 0,
where we used (9.3.7) and that E[φ s ] 2 Lip ≤ C with Lemma 9.3.2, where C does not depend on s. Furthermore, following (9.3.7) we have

E t 0 ⟨∇G((φ s ) n ), Z n 2 (•, s)⟩dW s - t 0 ⟨∇G(φ s ), Z 2 (•, s)⟩dW s 2 = t 0 E ⟨∇G((φ s ) n ), Z n 2 (•, s)⟩ -⟨∇G(φ s ), Z 2 (•, s)⟩ 2 ds ≤ 2∥∇G∥ ∞ ∥Z 2 ∥ ∞ t 0 E ⟨∇G((φ s ) n ), Z n 2 (•, s)⟩ -⟨∇G(φ s ), Z 2 (•, s)⟩ ds ≤ 2C∥∇G∥ ∞ ∥Z 2 ∥ ∞ ∥∇ 2 G∥ ∞ n -1 ∥Z 2 ∥ ∞ t 0 E[φ s ] Lip ds + ∥∇G∥ ∞ n -1 ∥∂ 1 Z 2 ∥ ∞ t 0
E∥φ s ∥ ∞ ds (9.3.10) and using Lemma 9.3.2, we have E∥φ s ∥ ∞ ≤ C and E[φ s ] Lip ≤ C where C does not depend on s, so that the quantity in (9.3.10) converges to 0 as n → ∞. Last, using (9.3.8) we get

E t 0 ⟨∇ 2 G((φ s ) n ), Z n 2 (•, s) ⊗2 ⟩ds - t 0 ⟨∇ 2 G(φ s ), Z 2 (•, s) ⊗2 ⟩ds ≤ C∥∇ 3 G∥ ∞ n -1 ∥Z 2 ∥ 2 ∞ t 0 E[φ s ] Lip ds + C∥∇ 2 G∥ ∞ ∥Z 2 ∥ ∞ ∥∂ 1 Z 2 ∥ ∞ n -1 t 0 E[∥φ s ∥ ∞ ]ds -→ n→∞ 0.

Proof of Theorem 9.2.2

In this section, we only give the full proof for (9.2.5). The proof of (9.2.6) is similar, as explained in Section 9.4.5.

Definition of the infinite dimensional semi-group and domino strategy

We define the infinite dimensional semi-group that we use for the domino strategy. We do not apply the domino strategy on X directly; instead we define an auxiliary process Y such that X can be induced from Y , as follows. Let us consider the following family of processes: where the interchange is ensured by the stochastic Fubini theorem. We also note that

Y t (u) = t 0 K 1 (t + u, s)b(X s )ds + t 0 K 2 (t + u, s)σ(X s )dW s , u ≥ 0, t ∈ [0, T ]. ( 9
Y t (0) = X t -X 0 , t ∈ [0, T ]. (9.4.3)
Then we have

X v = X 0 + Y t (v -t) + v t K 1 (v, s)b(X s )ds + v t K 2 (v, s)σ(X s )dW s , (9.4.4) Y v (u) = Y t (v -t + u) + v t K 1 (v + u, s)b(X s )ds + v t K 2 (v + u, s)σ(X s )dW s , (9.4.5) 0 ≤ t ≤ v ≤ T, u ≥ 0.
This leads us to define the following non-homogeneous semi-group P r,t for t ∈ [0, T ] and r ∈ [0, T -t] on Ω 2T :

P r,t (ω) u = ω r+u + t+r t K 1 (t + r + u, s)b( X s )ds + t+r t K 2 (t + r + u, s)σ( X s )dW s , u ≥ 0, (9.4.6)
where ( X s ) s∈ [t,t+r] is the solution of the following SVE:

X v = X 0 + ω v-t + v t K 1 (v, s)b( X s )ds + v t K 2 (v, s)σ( X s )dW s (9.4.7)
and where we omit here the dependency of X in t and in ω.

Since K i (u, s) = 0 for u ≥ 2T ,
Pr,t k (ω) u = ω r+u + t k +r t k K 1 (t k + r + u, s)b(X 0 + ω 0 )ds + t k +r t k K 2 (t k + r + u, s)σ(X 0 + ω 0 )dW s , (9.4.10) u ≥ 0, so that we have Ȳt (0) = Xt -X 0 , t ∈ [0, T ] (9.4.11)
and for r ∈ [0, h]:

Pr,t k ( Ȳt k ) u = Ȳt k (r + u) + t k +r t k K 1 (t k + r + u, s)b(X 0 + Ȳt k (0))ds + t k +r t k K 2 (t k + r + u, s)σ(X 0 + Ȳt k (0))dW s = Ȳt k +r (u).
By a slight abuse of notation, we use the notations P and P also to denote the infinitesimal generators such that for every g : Ω 2T → R, ω ∈ Ω 2T , t ∈ [0, T ] and r ∈ [0, T -t] we have P r,t g(ω) := Eg(P r,t (ω)), Pr,t g(ω) := Eg( Pr,t (ω)).

For general semi-groups Q 1 , . . . , Q r we denote their composition as

r k=1 Q k := Q 1 • . . . • Q r .
Then we obtain X T = Y T (0) + X 0 = P T,0 ( 0) 0 + X 0 and XT = N -1 k=0 Ph,t N -1-k ( 0) 0 + X 0 , where 0 denotes the path on R + constant to 0. Now for f : R → R being C 5 b we define

f : ω ∈ Ω 2T → f (ω 0 + X 0 ).
(9.4.12) Following Remark 9.3.1, we also have f ∈ C 5 b . Moreover we can write

Ef ( XT ) = Ef N -1 k=0 Ph,t N -1-k ( 0) 0 + X 0 = E f N -1 k=0 Ph,t N -1-k ( 0) = N -1 k=0 Ph,t k f ( 0).
We highlight that in our notations the order of the semi-groups is reversed whether k Ph,t k is applied to some ω ∈ Ω 2T or to some g : Ω 2T → R. We then rewrite the weak error as

Ef ( XT ) -Ef (X T ) = N -1 k=0 Ph,t k f ( 0) - N -1 k=0 P h,t k f ( 0) = N -1 k=0 P kh,0 • ( Ph,t k -P h,t k ) •   N -1 j=k+1 Ph,t j   f ( 0), (9.4.13)
where we used a telescopic sum.

Weak error in small time

In this section, we give a bound on the weak error in small time for the one-step Euler-Maruyama scheme ( Ph,t k -P h,t k )g(ω), where g : Ω 2T → R is some smooth functional and ω ∈ Ω 2T and ω ∈ C 2 .

Proposition 9.4.1. Let ω ∈ Ω 2T and be C 2 , g : Ω 2T → R with bounded (pathwise) derivatives up to order 5 and k ∈ {0, . . . , N -1}. Then we have

( Ph,t k -P h,t k )g(ω) ≤ C(1 + [ω] Lip )h 2 , (9.4.14)
where the constant C does not depend on k nor ω nor h.

Proof. We can assume that ω is Lipschitz-continuous without loss of generality.

• Let us consider ( X s ) s∈[t k ,t k +h] as defined in (9.4.7). Then for v ∈ [t k , t k+1 ] and ε ∈ [0, t k+1 -v] we have

X v+ε -X v = ω v+ε-t k -ω v-t k + v t k (K 1 (v + ε, s)-K 1 (v, s))b( X s )ds + v+ε v K 1 (v + ε, s)b( X s )ds + v t k (K 2 (v + ε, s) -K 2 (v, s))σ( X s )dW s + v+ε v K 2 (v + ε, s)σ( X s )dW s so that d X v = ωv-t k dv + K 1 (v, v)b( X v )dv + K 2 (v, v)σ( X v )dW v + v t k ∂ 1 K 1 (v, s)b( X s )ds + v t k ∂ 1 K 2 (v, s)σ( X s )dW s dv.
It follows from the classic Itō formula that for ψ : R → R and v ≥ t k we have

dψ( X v ) = ∇ψ( X v )(K 1 (v, v)b( X v )dv + ωv-t k dv + K 2 (v, v)σ( X v )dW v ) + 1 2 ∇ 2 ψ( X v )K 2 2 (v, v)σ 2 ( X v )dv + ∇ψ( X v ) v t k ∂ 1 K 1 (v, s)b( X s )ds + v t k ∂ 1 K 2 (v, s)σ( X s )dW s dv (9.4.15)
In particular we remark that for r ∈ [0, h] and if ψ is C 2 with bounded derivatives,

Eψ( X t k +r ) -ψ(X 0 + ω 0 ) ≤ ∥∇ψ∥ ∞ (∥K 1 ∥ ∞ ∥b∥ ∞ r + [ω] Lip ) + 1 2 ∥∇ 2 ψ∥ ∞ ∥K 2 ∥ 2 ∞ ∥σ∥ 2 ∞ r + ∥∇ψ∥ ∞ ∥∂ 1 K 1 ∥ ∞ ∥b∥ ∞ r 2 2 + ∥∇ψ∥ ∞ ∥∂ 1 K 2 ∥ ∞ ∥σ∥ ∞ 2r 3/2 3 ≤ C(1 + [ω] Lip )r, (9.4.16)
where we bound the last term as follows:

E t k +r t k ∇ψ( X v ) v t k ∂ 1 K 2 (v, s)σ( X s )dW s dv ≤ ∥∇ψ∥ ∞ t k +r t k E v t k ∂ 1 K 2 (v, s)σ( X s )dW s dv ≤ ∥∇ψ∥ ∞ ∥∂ 1 K 2 ∥ ∞ ∥σ∥ ∞ t k +r t k (v -t k ) 1/2 dv.
• On the other side for r ≥ 0 we have

P r+ε,t k (ω) u -P r,t k (ω) u = ω r+ε+u -ω r+u + t k +r+ε t k +r K 1 (t k + r + ε + u, s)b( X s )ds + t k +r+ε t k +r K 2 (t k + r + ε + u, s)σ( X s )dW s + t k +r t k (K 1 (t k + r + ε + u, s) -K 1 (t k + r + u, s))b( X s )ds + t k +r t k (K 2 (t k + r + ε + u, s) -K 2 (t k + r + u, s))σ( X s )dW s so that we can write dP r,t k (ω) u = ωr+u dr+K 1 (t k + r + u, t k + r)b( X t k +r )dr+K 2 (t k + r + u, t k + r)σ( X t k +r )dW t k +r + t k +r t k ∂ 1 K 1 (t k + r + u, s)b( X s )ds + t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s dr
so that for G : Ω 2T → R being C 3 b and using Theorem 9.3.4 (and checking the assumption (9.3.5) with Lemma 9.3.2) we obtain dG(P r,t k (ω)) = ⟨∇G(P r,t k (ω)), ωr+• ⟩dr + ⟨∇G(P r,t k (ω)), (K

1 (t k + r + u, t k + r)) u≥0 ⟩b( X t k +r )dr + ⟨∇G(P r,t k (ω)), (K 2 (t k + r + u, t k + r)) u≥0 ⟩σ( X t k +r )dW t k +r + 1 2 ⟨∇ 2 G(P r,t k (ω)), (K 2 (t k + r + u, t k + r)) ⊗2 u≥0 ⟩σ 2 ( X t k +r )dr + ∇G(P r,t k (ω)), t k +r t k ∂ 1 K 1 (t k + r + u, s)b( X s )ds + t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s u≥0
dr.

(9.4.17)

In particular, we remark that

EG(P r,t k (ω)) -G(ω) ≤ ∥∇G∥ ∞ [ω] Lip r + ∥∇G∥ ∞ ∥K 1 ∥ ∞ ∥b∥ ∞ r + 1 2 ∥∇ 2 G∥ ∞ ∥K 2 ∥ 2 ∞ ∥σ∥ 2 ∞ r + ∥∇G∥ ∞ ∥∂ 1 K 1 ∥ ∞ ∥b∥ ∞ r 2 2 + C∥∇G∥ ∞ ∥σ∥ ∞ ∥∂ 2 11 K 2 ∥ 2 ∞ 2r 3/2 3 , ≤ C(1 + [ω] Lip )r. (9.4.18)
where we used Lemma 9.3.2 to bound the last term. Thus for g : Ω 2T → R being C 5 b we have

Eg(P h,t k (ω)) -g(ω) = E h 0 ⟨∇g(P r,t k (ω)), ωr+• ⟩dr + 1 2 h 0 ⟨∇ 2 g(P r,t k (ω)), (K 2 (t k + r + u, t k + r)) ⊗2 u≥0 ⟩σ 2 ( X t k +r )dr + h 0 ⟨∇g(P r,t k (ω)), (K 1 (t k + r + u, t k + r)) u≥0 ⟩b( X t k +r )dr + h 0 ∇g(P r,t k (ω)), t k +r t k ∂ 1 K 1 (t k + r + u, s)b( X s )ds + t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s u≥0 dr =: 5 i=1 I i .
Likewise, we obtain a similar formula on Eg( Ph,t k (ω)) -g(ω), replacing b by b and σ by σ, and we write Eg( Ph,t k (ω)) -g(ω) =:

5 i=1
Īi .

We shall now inspect the quantity I i -Īi for every i = 1, . . . , 5.

• For fixed r ∈ [0, h], let G r : η → ⟨∇g(η), ωr+• ⟩. Then G r ∈ C 3 b and we have ⟨∇G r (η), τ ⟩ = ⟨∇ 2 g(η), ωr+• ⊗ τ ⟩, ⟨∇ 2 G r (η), τ 1 ⊗ τ 2 ⟩ = ⟨∇ 3 g(η), ωr+• ⊗ τ 1 ⊗ τ 2 ⟩.
Applying the Itō formula (9.4.17) again to α → EG r (P α,t k (ω)) for α ∈ [0, r] and with the estimate (9.4.18), we obtain

|EG r (P r,t k (ω)) -G r (ω)| ≤ Cr(1 + [ω] Lip ).
Similarly, we have

EG r ( Pr,t k (ω)) -G r (ω) ≤ Cr(1 + [ω] Lip ),
and then

I 1 -Ī1 = E h 0 G r (P r,t k (ω))dr- h 0 G r ( Ph,t k (ω))dr ≤ C(1+[ω] Lip ) h 0 rdr ≤ C(1+[ω] Lip )h 2 . • For fixed r ∈ [0, h], let G r : η → 1 2 ⟨∇ 2 g(η), (K 2 (t k + r + u, t k + r)) ⊗2 u≥0 ⟩.
Then G r ∈ C 3 b and we have

⟨∇G r (η), τ ⟩ = 1 2 ⟨∇ 3 g(η), (K 2 (t k + r + u, t k + r)) ⊗2 u≥0 ⊗ τ ⟩, ⟨∇ 2 G r (η), τ 1 ⊗ τ 2 ⟩ = 1 2 ⟨∇ 4 g(η), (K 2 (t k + r + u, t k + r)) ⊗2 u≥0 ⊗ τ 1 ⊗ τ 2 ⟩.
Applying the Itō formulae we obtained in (9.4.17) and in (9.4.15) and the classic Itō formula for a product, we get

E[G r (P r,t k (ω))σ 2 ( X t k +r )] -G r (ω)σ 2 (X 0 + ω 0 ) = E r 0 d G r (P α,t k (ω)) σ 2 ( X t k +α ) + G r (P α,t k (ω))d σ 2 ( X t k +α ) + d⟨G r (P •,t k (ω)), σ 2 ( X t k +• )⟩ α =: A 1 + A 2 + A 3 ,
but σ 2 is bounded and following (9.4.18), we obtain that A 1 ≤ C(1 + [ω] Lip )r; the same way and since G r is bounded (independently on r) and following (9.4.16) we have

A 2 ≤ C(1 + [ω] Lip )r. Moreover we have d⟨G r (P •,t k (ω)), σ 2 ( X t k +• )⟩ α = ∇σ 2 ( X t k +α )K 2 (t k + α, t k + α)σ 2 ( X t k +α )
• ⟨∇G r (P α,t k (ω)), (K 2 (t k + α + u, t k + α)) u≥0 ⟩dα so that same way we get A 3 ≤ Cr. Thus we finally obtain

EG r (P r,t k (ω))σ 2 ( X t k +r ) -G r (ω)σ 2 (X 0 + ω 0 ) ≤ C(1 + [ω] Lip )r. (9.4.19)
The same way we have

EG r ( Pr,t k (ω))σ 2 (X 0 + ω 0 ) -G r (ω)σ 2 (X 0 + ω 0 ) ≤ C(1 + [ω] Lip )r, so that
I 2 -Ī2 = E h 0 (G r (P r,t k (ω))σ 2 ( X t k +r ) -G r ( Pr,t k (ω))σ 2 (X 0 + ω 0 ))dr ≤ C(1 + [ω] Lip )h 2 .
• For r ∈ [0, h] and u ≥ 0 let us define

φ r u := t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s
and let us write

dφ r u = ∂ 1 K 2 (t k + r + u, t k + r)σ( X t k +r )dW t k +r + t k +r t k ∂ 2 11 K 2 (t k + r + u, s)σ( X s )dW s dr. (9.4.20)
Moreover, using Lemma 9.3.2, we have E∥φ r ∥ 2 ∞ ≤ C. We also define

G : (η 1 , η 2 ) ∈ Ω 2 2T → ⟨∇g(η 1 ), η 2 ⟩. Then G ∈ C 3 b with ⟨∇G(η 1 , η 2 ), (τ 1 , τ 2 )⟩ = ⟨∇ 2 g(η 1 ), τ 1 ⊗ η 2 ⟩ + ⟨∇g(η 1 ), τ 2 ⟩, ⟨∇ 2 G(η 1 , η 2 ), (τ 1 , τ 2 ) ⊗2 ⟩ = ⟨∇ 3 g(η 1 ), (τ 1 ) ⊗2 ⊗ η 2 ⟩ + 2⟨∇ 2 g(η 1 ), τ 1 ⊗ τ 2 ⟩.
Using (9.4.17) and (9.4.20), for every r ∈ [0, h] we have ⟨∇g(P r,t k (ω)),

t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s u≥0 ⟩ = r 0 ⟨∇ 2 g(P α,t k (ω)), ωr+• ⊗ φ α ⟩ + ⟨∇ 2 g(P α,t k (ω)), (K 1 (t k + α + u, t k + α)) u≥0 ⊗ φ α ⟩b( X t k +α ) + 1 2 ⟨∇ 3 g(P α,t k (ω)), (K 2 (t k + α + u, t k + u)) ⊗2 u≥0 ⊗ φ α ⟩σ 2 ( X t k +α ) + ⟨∇ 2 g(P α,t k (ω)), t k +α t k ∂ 1 K 1 (t k + α + u, s)b( X s )ds + t k +α t k ∂ 1 K 2 (t k + α + u, s)σ( X s )dW s u≥0 ⊗ φ α ⟩ dα + r 0 ⟨∇ 2 g(P α,t k (ω)), (K 2 (t k + α + u, t k + u)) u≥0 ⊗ φ α ⟩σ( X t k +α )dW t k +α + r 0 ⟨∇g(P α,t k (ω)), t k +α t k ∂ 2 11 K 2 (t k + α + u, s)σ( X s )dW s u≥0 ⟩dα + r 0 ⟨∇g(P α,t k (ω)), (∂ 1 K 2 (t k + α + u, t k + α)) u≥0 ⟩σ( X t k +α )dW t k +α + r 0 ⟨∇ 2 g(P α,t k (ω)), (K 2 (t k + α + u, t k + u)) u≥0 ⊗ (∂ 1 K 2 (t k + α + u, t k + α)) u≥0 ⟩ • σ 2 ( X t k +α )dα so that
E⟨∇g(P r,t k (ω)),

t k +r t k ∂ 1 K 2 (t k + r + u, s)σ( X s )dW s u≥0 ⟩ ≤ C(1 + [ω] Lip )r.
The same way, we obtain E⟨∇g( Pr,t k (ω)),

t k +r t k ∂ 1 K 2 (t k + r + u, s)σ(X 0 + ω(0))dW s u≥0 ⟩ ≤ C(1 + [ω] Lip )r and then |I 5 -Ī5 | ≤ C(1 + [ω] Lip )h 2 .
• The arguments to prove that

|I 3 -Ī3 | + |I 4 -Ī4 | ≤ C(1 + [ω] Lip )h 2
are the same or more simple.

Proof that the derivatives of g are bounded

In this section, we prove that if we choose g : Ω → R as in (9.4.13), then g has bounded derivatives up to order 5 so that we can apply Proposition 9.4.1 to g.

Lemma 9.4.2. Let k ∈ {0, . . . , N -1}, let us define

g(ω) := N -1 j=k Ph,t j f (ω) = E   f ( N -1 j=k Ph,t N -1+k-j • ω) 0   ,
where f is defined in (9.4.12). Then g is five times differentiable with

∥∇ ℓ g∥ ∞ ≤ C, ℓ ∈ {1, . . . , 5}.
Proof. We can rewrite

g(ω) = E f (ω T -t k + T t k K 1 (T, s) b( Xω s )ds + T t k K 2 (T, s)σ( Xω s )dW s )
where Xω follows the piecewise SVE:

Xω v = X 0 + ω v-t k + j ℓ=k b( Xω t ℓ ) t ℓ+1 ∧v t ℓ K 1 (v, s)ds + j ℓ=k σ( Xω t ℓ ) t ℓ+1 ∧v t ℓ K 2 (v, s)dW s , v ∈ [t j , t j+1 ], j ∈ {k, . . . , N -1}.
The process Xω depends on k, but we omit this dependency in the notation without ambiguity. We define the tangent process (Z ω v ) v∈[T -t k ,T ] as the process of the linear operators on Ω 2T by induction as:

⟨Z ω v , η⟩ = η v-t k + j ℓ=k ∇b( Xω t ℓ ) • ⟨Z ω t ℓ , η⟩ t ℓ+1 ∧v t ℓ K 1 (v,
⟨Z ω v , η⟩ = η v-t k + v t k K 1 (v, s)∇ b( Xω s )⟨Z ω s , η⟩ds + v t k K 2 (v, s)∇σ( Xω s )⟨Z ω s , η⟩dW s . Let us denote φ v := sup s∈[t k ,v] E∥Z ω s ∥ 2 , v ∈ [t k , T ]
and we have (using the inequality (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 )):

φ v ≤ 3 + 3 v t k (T ∥K 1 ∥ 2 ∞ ∥∇b∥ 2 ∞ + ∥K 2 ∥ 2 ∞ ∥∇σ∥ 2 ∞ )φ s ds
so that using the Gronwall inequality:

φ v ≤ 3 exp (v -t k )(T ∥K 1 ∥ 2 ∞ ∥∇b∥ 2 ∞ + ∥K 2 ∥ 2 ∞ ∥∇σ∥ 2 ∞ ) ≤ C.
Then we have

⟨∇g(ω), η⟩ = E   ∇f ( N -1 j=k Ph,t N -1+k-j • ω) 0 • η T -t k + T t k K 1 (T, s)∇ b( Xω s )⟨Z ω s , η⟩ds + T t k K 2 (T, s)∇σ( Xω s )⟨Z ω s , η⟩dW s implying ∥∇g∥ ∞ ≤ ∥∇f ∥ ∞ 1 + T t k ∥K 1 ∥ ∞ ∥∇b∥ ∞ φ 1/2 s ds + ( T t k ∥K 2 ∥ 2 ∞ ∥∇σ∥ 2 ∞ φ s ds) 1/2
thus implying that ∇g is bounded (independently of k and N ).

We prove that the derivatives of g are bounded up to order 5 by following the same method.

Conclusion: proof of Theorem 9.2.2

Proof. Let us consider (9.4.13) again and for k ∈ {0, . . . , N -1} we set ω k := P kh,0 ( 0) and

g k+1 := N -1 j=k+1
Ph,t j f .

Then following Lemma 9.4.2, we have that g k+1 ∈ C 5 b . On the other side, we have that

ω k is C 2 with ωk u = t k 0 ∂ 1 K 1 (t k + u, s)b( X s )ds + t k 0 ∂ 1 K 2 (t k + u, s)σ( X s )dW s , ωk u = t k 0 ∂ 2 11 K 1 (t k + u, s)b( X s )ds + t k 0 ∂ 2 11 K 2 (t k + u, s)σ( X s )dW s
where the interchange is ensured by the stochastic Fubini theorem, and following Lemma 9.3.2, we obtain that E[ω k ] Lip ≤ C. Then applying Proposition 9.4.1 with g = g k+1 and ω = ω k we get

P kh,0 • ( Ph,t k -P h,t k ) •   n-1 j=k+1 Ph,t j   f ( 0) ≤ Ch 2 . Summing over k ∈ {0, . . . , N -1} yields E[f ( XT )] -E[f (X T )] = O 1 N .

Proof of weak error for the scheme with discretization of the kernels

The proof of (9.2.6) for the scheme ⇀ X defined in (9.2.4) is similar to the proof for (9.2.5). We define the associated semi-group as

⇀ P r,t k (ω) u = ω r+u + t k +r t k K 1 (t k + r + u, t k )b(X 0 + ω 0 )ds 10 1 10 2 10 -4 10 -3 N Ef ( ⇀ S ⌊βN ⌋ T ) - Ef ( ⇀ S N T )
log-log linear regression, slope ≃ -0.972 Figure 9.2: Simulation of (9.5.1) and weak error in log-log scale. We give in red the 95% confidence intervals where the number of trajectories is 512 × 1024 × 2000 for each value of N .

Then there exists a modification X of X which is α-Hölder continuous for every α ∈ (0, ε/p) and with

E sup t,s∈[0,T ], t̸ =s | X t -X s | |t -s| α p ≤ C 0 2 1+α 1 -2 -α p T 1+ε-αp 1 -2 -(ε-αp) .
Proof. We refer to the proof of [LG16, Theorem 2.9, Lemma 2.10], with an immediate adaptation if we do not assume T = 1.

Introduction

The objective of the present article is to provide exact formulas and algorithms for the simulation of a normally reflected two-dimensional Brownian motion starting at x 0 ∈ R 2 .

The reflected process is denoted by X ≡ (X t ) t∈[0,T ] and the domain of reflection is a wedge D, i.e. a subset of R 2 delimited by two (non parallel) lines, so as to give Monte Carlo methods for the estimation of E x 0 [f (X T )], where f : D → R is a measurable function such that f (X T ) ∈ L 1 and T > 0 is the finite time horizon. Our first goal is to obtain an explicit formula for the density of X t (see Theorem 10.4.2). Unfortunately, this formula involves oscillating infinite sums of Bessel functions. Instead of directly computing these sums, we propose an alternative simulation method which uses an extension of the reflection principle in two dimensions for a particular type of wedges with angle π/m, m ∈ N. As a first step, we obtain a simulation method for E x 0 [f (W T ∧τ )], where W = (W t ) t≥0 is a two-dimensional Brownian motion and τ stands for the first time the process W touches the boundary of the wedge D. Applying the methodology for the stopped process recursively one obtains an algorithm for the reflected process X. The algorithms we present here rely on an idea which has links to the rectangle method [DL06]: we include a smaller wedge of angle π/m inside the first wedge and simulate the exit from the smaller wedge.

The algorithm for the simulation of the stopped process improves the method proposed in [START_REF] Metzler | Multivariate first-passage models in credit risk[END_REF]. In the reflected case, we prove that although some lower moments are finite, the average number of simulations for this method is infinite. This effect is created due to the large number of reflections that may happen if the process enters the corner of the wedge. We then propose a modified approximation scheme which takes into account the asymptotic behaviour of the density near the corner. With this modification, we show that the expected number of iterations of the algorithm is finite and measure the error of approximation in total variation distance. We then give adaptations of our algorithms to more general Itō processes reflected in a wedge.

The expression of the density of the stopped Brownian motion in a wedge of R 2 has been obtained in [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF] using the method of images. However, the author does not provide full arguments for the verification of the initial condition and some exchanges of infinite summation and integrations are not fully explained. For the history of the expression of the density of the stopped Brownian motion on a wedge, see [START_REF] Bañuelos | Brownian motion in cones[END_REF].

In [START_REF] Chupeau | Survival probability of a Brownian motion in a planar wedge of arbitrary angle[END_REF] and [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF] the authors correct some mistakes in formulas appearing in [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF] and give formulas for a number of other random variables related to the stopped Brownian motion in two dimensions with general correlation coefficients, such as the survival probability and the first-passage time distribution. However, formulas from [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF] and [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF] are not directly applicable to simulation algorithms of stopped processes, as they involve infinite sums and Bessel functions. [Met09, Section 2] gives an approximate simulation algorithm using the semi-analytic expression of the density established in [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF]. But this method implies the approximation of an infinite sum of Bessel functions and it leads to bias when simulating the stopping time τ .

More recent papers ([EFW13], [START_REF] Blanchet-Scalliet | Hitting time for correlated three-dimensional Brownian motion[END_REF]) extend the results of [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF] and [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF] to three dimensions and give algorithms and density formulas for the stopped Brownian motion in three dimensions, using the method of images in R 3 . These algorithms which point towards the possibility of larger dimension extensions are however limited to some specific values of correlation. In [KLR18], the authors use the spectral decomposition method which applies in a larger generality but still relies on efficient computational methods for eigenvalue calculations and truncations of infinite sums. These references concentrate on the stopped Brownian motion case and there is no discussion about the simulation of reflected Brownian motion in wedges.

Simulation algorithms for the reflected Brownian motion have been only partly studied, although they share similarities with the stopped case. Simulation algorithms for quantities related to reflected Brownian motion on an orthant in multi-dimensions can be found in [START_REF] Blanchet | Exact simulation of multidimensional reflected Brownian motion[END_REF] and [START_REF] Blanchet | Steady-state simulation of reflected Brownian motion and related stochastic networks[END_REF]. They do not use the method of images, but the characterization through the Skorokhod problem and the so-called ε-strong methodology which relies on the regularity of the reflection functional. Still, the method proposed has a theoretically infinite mean running time.

Other approximations methods which are closer to random walk versions of the reflected Brownian motion on tile domains are studied in [START_REF] Dubédat | Reflected planar Brownian motions, intertwining relations and crossing probabilities[END_REF] and [START_REF] Kager | Reflected Brownian motion in generic triangles and wedges[END_REF]. Furthermore, other simulations methods for reflected Brownian motion which do not cover the case of the wedge can be found in [START_REF] Costantini | Numerical approximation for functionals of reflecting diffusion processes[END_REF], [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF], [START_REF] Burdzy | Discrete approximations to reflected Brownian motion[END_REF] and [START_REF] Bayer | Adaptive weak approximation of reflected and stopped diffusions[END_REF] the references therein.

The reflected Brownian motion and more generally reflected processes have applications in finance (for example, in stochastic models where the process, which can model an interest rates, is constrained to be non-negative [START_REF] Ha | Applications of the reflected Ornstein-Uhlenbeck process[END_REF] or has other constraints like barriers such as in [START_REF] Ichiba | Diffusions with rankbased characteristics and values in the nonnegative quadrant[END_REF]); in queueing models [START_REF] Giorno | On some diffusion approximations to queueing systems[END_REF], etc. In particular, in the recent years there has been a lot of developments concerning the study of stationary measures of reflected Brownian motion such as [START_REF] Dieker | Reflected Brownian motion in a wedge: sum-ofexponential stationary densities[END_REF] which is slightly related to the ideas which we use here to construct a simulation method on a wedge. For a review, on this topic and the relationship with queuing theory see e.g. [START_REF] Dieker | Reflected Brownian motion[END_REF].

Considering the particular case of a wedge may open the way to new simulation algorithms for reflected processes adapted to non-smooth domains with corner points in two or more dimensions (see Section 10.11).

The paper is organized as follows. In Section 10.3 we state the framework of the problem, giving a parametrization of the wedge in R 2 . We also give the formula for the density of the stopped Brownian motion in a wedge of angle α ∈ (0, 2π). The density of the stopped Brownian motion had already been obtained in [START_REF] Iyengar | Hitting lines with two-dimensional Brownian motion[END_REF]. A more general case is studied in [START_REF] Bañuelos | Brownian motion in cones[END_REF] which provides a detailed history and a full proof that the density formula is the fundamental solution of the associated partial differential equation.

In Section 10.4, we provide the density of the reflected Brownian motion on the wedge. First, we use the method of images in a special type of wedges as it serves to understand how one induces the density of the reflected Brownian motion in the general case. In the Appendix we include the proof in the reflected case on more general cones following [START_REF] Bañuelos | Brownian motion in cones[END_REF].

In Section 10.5, we give explicit algorithms for the simulation of the stopped and reflected Brownian motions. To do so, we first derive from [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF] and the formulas proved in Section 10.4 expressions of densities of the exit time and exit point from the wedge. Then, we give an algorithm for the simulation of the stopped Brownian motion using the explicit expressions in the case α = π/m. The simulation algorithm is based on the simulation of a random sequence of stopped Brownian motion processes on domains with angle α = π/m. This first algorithm is useful for the simulation of reflected Brownian motion, but it also has its own interest. Then we give the simulation algorithm for the reflected Brownian motion with unit diffusion matrix.

In Section 10.6, we study the mean number of iterations of the algorithm to finish, denoted

E[N ].
For the stopped Brownian motion, E[N ] is bounded above by a constant. The reflected case is more difficult to deal with. In this case, N is equal to the number of times a Brownian motion goes from one frontier of a wedge to another. We first give theoretical bounds for E[N ] and prove that although the number of iterations is almost surely finite, E[N ] is infinite, which is due to the events when the Brownian motion comes close to the origin. We then propose a modification of the algorithm: we stop the algorithm when the Brownian motion comes too close to the origin, measured by a parameter ε and use an approximation of the density for the reflected Brownian motion close to the origin, induced by the density formula in Section 10.4. We then prove that for the modified algorithm, E[N ] has an upper bound which grows as ε 1-p for all p ∈ (1, 2), and that the error of approximation is of order ε. In Section 10.7, we show how the algorithms can directly be applied for the simulation of the reflected and stopped Brownian motions with non-zero drift. Then we give an algorithm for the simulation of a class of Itō diffusion processes, reflected or stopped in a wedge in R 2 . This algorithm is in fact a direct application of the precedent algorithms, combined with a Euler-Maruyama scheme.

Finally, in Section 10.8, we perform Monte Carlo simulations for the estimation of reflected and stopped Brownian motions and Itō processes. We show that the bias in the algorithm from [START_REF] Metzler | Multivariate first-passage models in credit risk[END_REF] can be significant compared with our method which is exact. Simulations prove the need for the approximation algorithm for the reflected Brownian motion, as the exact algorithm takes too much time to be efficient. Clearly, one remaining subject which is not considered here is the reflecting case with non-unit diffusion coefficient. This remains an open problem.

The authors would like to express thanks to the anonymous referee which provided references [START_REF] Bañuelos | Brownian motion in cones[END_REF] and [START_REF] Dieker | Reflected Brownian motion in a wedge: sum-ofexponential stationary densities[END_REF] which improved the results and shortened the proof of Theorem 10.4.2.

Notations

We give a brief list of notations that are used through the text.

The natural logarithm is denoted with log. We frequently use the notation ± in order to denote the two rays that define a wedge. Sometimes rather than writing two equations we just write one using the symbol ± or ∓ meaning that we are stating two equations, one using the top symbols and another using the bottom symbols that appear throughout the equation.

I a stands for the modified Bessel function of the first kind given for a ≥ 0, x ≥ 0 by:

I a (x) = ∞ k=0 x a+2k 2 a+2k k!Γ(k + a + 1) . (10.2.1)
This function satisfies the differential equation Now we define the probabilistic setting as follows. Let (Ω, F, P, (F t ) t≥0 ) be a filtered probability space satisfying the usual conditions. Let T > 0 be the time horizon and let W = (W t ) t≥0 be a two-dimensional correlated Brownian motion on (Ω, F, P, (F t ) t≥0 ). We denote by Σ the covariance matrix of W and we assume that Σ is a non-singular matrix. We define a family of probabilities on (Ω, F, (F t ) t≥0 ) by: ∀x 0 ∈ R 2 , P x 0 := P |W 0 =x 0 .

x 2 d 2 I a (x) dx 2 + x dI a (x) dx -(x 2 + a 2 )I a (x) = 0, x ≥ 0. ( 10
We will denote by E x 0 the expectation under P x 0 .

If W = (W t ) t≥0 is a Brownian motion we will say "stopped Brownian motion" to designate the process (W t∧τ ) t≥0 , and "killed Brownian motion" to designate the process (W t 1 τ >T ) t∈[0,T ] , where τ is some stopping time defined later.

We will frequently use in the proofs the Bessel process (R t ) t≥0 in dimension δ ≥ 2, i.e. a real-valued process satisfying the stochastic differential equation:

R t = r 0 + B t + t 0 δ -1 2 ds R s ,
where (B t ) t≥0 is a standard one-dimensional Brownian motion starting from zero. The index of the Bessel process is defined as ν = δ 2 -1. For a general reference to these matters we refer to [JYC09, Section 6].

Setting of the problem

Let a ∈ R ∪ {±} and consider the subset D ⊂ R 2 defined by one of the four following cartesian equations:

D =          {(x, y) ∈ R 2 ,
y ≥ 0 and y ≤ ax} with a > 0, {(x, y) ∈ R 2 , y ≥ 0 and y ≥ ax} with a < 0, {(x, y) ∈ R 2 , y ≥ 0 or y ≥ ax} with a > 0, {(x, y) ∈ R 2 , y ≥ 0 or y ≤ ax} with a < 0.

(10.3.1)

The set D is called a wedge1 . The angle of the wedge is denoted by α and chosen such that α ∈ (0, 2π). An example is given in Figure 10.1. We write the wedge D in polar coordinates as follows:

D = {(r cos(θ), r sin(θ)), r ∈ R + , θ ∈ [0, α]}.
We define its boundary as ∂D = ∂D -∪ ∂D + , where In general, we use the notation D = ⟨α⟩ to denote a wedge which boundary is determined by the rays ∂D ± . In a similar way, we also generalize this notation to a general wedge which boundaries are determined by the rays at the angles 0 ≤ α < β < 2π as V = ⟨α, β⟩.

We use the following stopping times:

τ = inf {t > 0, W t ∈ ∂D}. τ ± = inf {t > 0, W t ∈ ∂D ± }.
As in [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF], we parametrize the covariance matrix of W via its Cholesky decomposition and write Σ = σσ T with

σ = σ 1 1 -ρ 2 σ 1 ρ 0 σ 2 ,
where ρ ∈ (-1, 1) and σ 1 , σ 2 ≥ 0. Then assuming that σ is invertible, we consider the process W ′ := σ -1 W , which is a standard two-dimensional Brownian motion with independent components. Furthermore, using the explicit formula for the inverse matrix, we deduce that

W 2 = 0 ⇐⇒ W ′2 = 0,
2 = aW 1 ⇐⇒    W ′2 = aσ 1 √ 1-ρ 2 σ 2 -aσ 1 ρ W ′1 if σ 2 -aσ 1 ρ ̸ = 0, W ′1 = 0 if σ 2 -aσ 1 ρ = 0.
In the case that σ 2 -aσ 1 ρ = 0, then the problem reduces to two independent one-dimensional Brownian motions being reflected or stopped in the first quadrant, and therefore the problem can be reduced to using two independent reflected or stopped Brownian motions in one dimension. For explicit formulas, see (10.4.2) and (10.4.1). Thus in the following we assume that σ 2 -aσ 1 ρ ̸ = 0. So τ -is the first passage time of W ′ through the horizontal axis and τ + is the time of the first passage of W ′ through the line y = a ′ x, where

a ′ = aσ 1 1 -ρ 2 σ 2 -aσ 1 ρ , ( 10
.3.2) so that τ = τ -∧ τ + is the exit time of W ′ from the wedge D ′ = ⟨α ′ ⟩, where α ′ is defined as arctan(a ′ ) or π + arctan(a ′ ) or 2π + arctan(a) depending on the cases. The process W ′ starts at x ′ 0 = σ -1 • x 0 . The possible cases are summed up in Table 10.2. After the above transformation the original wedge is transformed into a new one according to this table. Therefore mutatis-mutandis, we shall omit the primes in the notation and without loss of generality we directly assume that the process W has the correlation matrix Σ = I 2 and the wedge is given as in (10.3.1) with angle α.

In some results, we assume that the angle of the wedge satisfies the condition:

α = π m ,
for some m ∈ N. We define adjacent wedges as follows. For k = 0, 1, ..., 2m -1, let

D k := {(r cos(θ), r sin(θ)) : r ∈ [0, +∞), θ ∈ [kα, (k + 1)α]}.
We remark that D 0 = D and that R 2 = 2m-1 k=0 D k . We also define the transformation T k : From the definition, it follows that T k is an isometric bijection between D 0 and D k . This setting is represented in Figure 10.3.

T k :D 0 -→ D k T k ((
The density of the killed Brownian motion and some related random quantities appear in [START_REF] Metzler | On the first passage problem for correlated Brownian motion[END_REF]. We first quote the explicit density result for killed Brownian motion which is fully proven in [BnS97, Lemma 1] for generalized multidimensional cones.

Theorem 10.3.1 (Killed case). Assume that D = ⟨π/m⟩ for some m ∈ N. Then we have the following formula for the density of killed Brownian motion on the wedge D:

∀t > 0, x, y ∈ D, P x (W t ∈ dy, τ > t) = 1 2πt 2m-1 k=0 (-1) k e -|x-T k y| 2 2t dy. (10.3.3)
In general, let α ∈ (0, 2π), D = ⟨α⟩ and x = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) ∈ D. Then for any y = (r cos(θ), r sin(θ)) ∈ D, the density of the killed Brownian motion on the wedge D is given by

P x (W t ∈

Analytic formulas for the density of the reflected process

In this section, we give explicit expressions for the densities of the reflected Brownian motion. Before doing this, we review the one dimensional case which will also help explain the motivation for the simulation methods to be introduced later. The one-dimensional case can be treated using the classic reflection principle, which directly gives the density of the killed and reflected Brownian motions. In the one-dimensional case, we can assume that D = [0, +∞) and that Σ = 1. Then for all non-negative measurable functions f : D → R and x 0 ≥ 0,

E x 0 [f (W T )1 τ >T ] = E x 0 f (W T )1 W T >0 1 -e -x 0 (x 0 +W T ) T , ( 10
.4.1)

E x 0 [f (X T )] = E x 0 f (W T )1 W T >0 1 + e -x 0 (x 0 +W T ) T . (10.4.2)
The above formulas show that the simulation of the killed and reflected Brownian motions in the one-dimensional case can be easily performed using changes of measures on the original unrestricted Brownian motion.

The proof in the reflected case in R 2 shares common arguments with the killed case. Throughout these arguments, one uses results from classical theory of partial differential equations on wedges. For a general reference on this topic, we refer the reader to [LSU68, chapter IV] and [START_REF] Kou | First-passage times of two-dimensional Brownian motion[END_REF] where one can find existence and uniqueness results for the partial differential equations that are related to the problems in this article. Definition 10.4.1. We define the normal reflection process of a continuous adapted stochastic process ξ on the subset D, as the unique solution (X t , L t ) t≥0 of the following problem (see for example [START_REF] Pilipenko | An introduction to stochastic differential equations with reflection[END_REF]):

1. X t = ξ t + L t , where L = (L t ) t≥0 is some adapted process.

2. ∀t ≥ 0, X t ∈ D.

L is a continuous process of bounded variation and L

0 = 0. 4. ∀t ≥ 0, L t = t
0 n(X s )1 Xs∈∂D d|L| s , which means that the process L increases only when X t ∈ ∂D and that the reflection is normal to the boundary.

Therefore the normally reflected Brownian motion is the process that is obtained from the above equation by taking ξ = W . In the one dimensional case, L corresponds to the local time of Brownian motion.

The existence and uniqueness of a Brownian motion reflected in a wedge, which is a nonsmooth domain, has been proved in the decorrelated case in [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF]. In this article we will only consider normal reflections in the case of unitary diffusion matrix. Note that if we use the decorrelation step as in (10.3.2) for the reflected Brownian motion, this transformation changes the normal reflected process into an obliquely reflected process. We do not know how to obtain the density of such a process. In the general case for α ∈ (0, 2π), D = ⟨α⟩ and x = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) ∈ D we have that for any y = (r cos(θ), r sin(θ)) ∈ D, the density of the reflected Brownian motion on the wedge D = ⟨α⟩ is given by

P x (X t ∈ dy) = 2r tα e - r 2 +r 2 0 2t 1 2 I 0 rr 0 t + ∞ n=1 I nπ/α rr 0 t cos nπθ α cos nπθ 0 α drdθ.
(10.4.4)

The proof of the above theorem is given in 10.10. The idea of using the simple case D = ⟨π/m⟩ to infer a general result appears in the literature of the image method. It has been used in a non-trivial way in order to deduce stationary measures for reflected processes (see e.g. [START_REF] Dieker | Reflected Brownian motion in a wedge: sum-ofexponential stationary densities[END_REF]). A formula similar to (10.4.4) appears in [CJ59, page 379 (8)], which is proved using Laplace transform inversion methods. We have decided to include a proof which uses the image method because it gives some intuition on the simulations methods to be introduced later.

Exact simulation algorithms

Theorems 10.3.1 and 10.4.2 give formulas that can be directly used to simulate the final values of the reflected and stopped processes in the case that the wedge angle is α = π/m (see Sections 10.5.2 and 10.5.3). However, direct algorithms that arise due to these results demand the use of approximations and larger computational time in general. Instead, we investigate an alternative simulation method using the tractable case α = π/m and give algorithms for the exact simulation of the reflected and stopped processes in any wedge. 10.5.1 Formulas for the simulation of (τ, W τ ) in the case α = π/m Before providing the simulation method, we give a formula that will be the key to provide a simplified simulation method which avoids the calculation of Bessel functions.

Theorem 10.5.1. Let V = ⟨α -, α + ⟩ and assume that α + -α -= π/m =: α for some m ∈ N and let τ be the hitting time on the wedge V for W which starts at x 0 = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) ∈ V, then where y ± = (r cos(α ± ), r sin(α ± )) ∈ ∂V ± . We use the notation

P x 0 (τ ∈ dt, W τ ∈ dy ± ) = r 0 2πt 2 e -
γ ± k := ± α ± ± 2kπ m ∓ θ 0 . (10.5.2)
Proof. We can assume that α -= 0 without loss of generality. We use the formulas [Met10, (1.5), (1.6)]. If y ∈ ∂D -, writing y = (r, 0) for some r > 0, we have:

P x 0 (τ ∈ dt, W τ ∈ dy) = π α 2 tr e - r 2 +r 2 0 2t ∞ n=1 n sin nπθ 0 α I nπ/α rr 0 t drdt.
(10.5.3) Using (10.3.4) and switching the order of derivation and integration, which can be justified (see the proof of Theorem 10.4.2 with Corollary 10.10.1 in the Appendix), we obtain for y ∈ ∂D -:

P x 0 (τ ∈ dt, W τ ∈ dy) = 1 2r 2 ∂ ∂θ θ=0 P x 0 (τ > t, W t ∈ d(r cos(θ), r sin(θ))).
So that in the case α = π/m, using (10.3.3) in polar coordinates and the definition of ϑ k : (10.5.4)

P x 0 (τ ∈ dt, W τ ∈ dy) = 1 2r 2 ∂ ∂θ θ=0 r 2πt e - r 2 +r 2 0 2t 2m-1 k=0 (-1) k e 1 t rr 0 cos(θ 0 -ϑ k ) drdt = r 0 2πt 2 e -
Likewise, we obtain a similar formula for the case y ∈ ∂D + .

From here, we present a method to simulate τ and W τ in the case of the wedge V = ⟨α -, α + ⟩ with α + -α -= π/m. For the simulation of W τ , we use the same method which is proposed in [Met09, Section 2.1] but we restrict to the case α = π/m. On the other hand, we propose an unbiased simulation method for τ in comparison with the method proposed in [START_REF] Metzler | Multivariate first-passage models in credit risk[END_REF] which has a bias difficult to estimate (see [Met09, Proposition 2.1.8] and the remark that follows).

Simulation of W τ : First, we simulate the selection of the boundary line on which W τ arrives, with a Bernoulli distribution. These two boundaries are denoted by ∂V ± = {(r cos(α ± ), r sin(α ± )) : r ≥ 0}. Following [Met09, Corollary 2.1.5] we have:

P x 0 (W τ ∈ V + ) = θ 0 -α - α + -α -, (10.5.5)
We simulate on which frontier W τ arrives first and then we directly simulate W τ , without simulating τ yet. Following [Met09, Proposition 2.1.6] one can compute explicit formulas for the distribution function of the radius of exit given that it exits at V + or V -. Also its inverse can be computed and therefore the inverse transformation method for simulation can be applied. Using these formulas, we can simulate the radius r τ as:

r τ =          r 0 cos πθ 0 α - sin πθ 0 α tan((π-πθ 0 /α)(U -1)) α/π if W τ ∈ V -, r 0 -cos πθ 0 α - sin πθ 0 α tan((πθ 0 /α)(U -1)) α/π if W τ ∈ V + , (10.5.6) with U ∼ U([0, 1]).
Simulation of τ : Knowing W τ , we simulate τ according to the conditional formula (see (10.5.4) and recall that dy ± = (dr cos(α ± ), dr sin(α ± ))):

P x 0 (τ ∈ dt| W τ = y ± ) = (P x 0 (W τ ∈ dy ± )) -1 dr r 0 2πt 2 m-1 k=0 sin(γ ± k ) • exp - (r -r 0 cos(γ ± k )) 2 + r 2 0 sin 2 (γ ± k ) 2t
dt.

(10.5.7)

Since some of the coefficients sin(γ ± k ) are negative, we cannot directly simulate according to this distribution as a mixture. Instead we select the indexes k ± 0 , . . . , k ± p ± such that for all i, sin(γ ± k ± i ) ≥ 0 and write:

P x 0 (τ ∈ dt| W τ = y ± ) ≤ (P x 0 (W τ ∈ dy ± )) -1 dr r 0 2πt 2 p ± i=0 sin(γ ± k ± i ) • exp   - (r -r 0 cos(γ ± k ± i )) 2 + r 2 0 sin 2 (γ ± k ± i ) 2t   dt. (10.5.8)
This is the distribution of a discrete mixture of random variables ξ k for 0 ≤ k ≤ p ± , where ξ k is the inverse of an exponential random variable of parameter ((r -r 0 cos(γ ±

k ± i )) 2 + r 2 0 sin 2 (γ ± k ± i ))/2.
With this information, we build an acceptance-rejection sampling, based on the density on the right hand side of (10.5.8) and accept the sample value t under the condition

U < m-1 k=0 sin(γ ± k ) exp - (r -r 0 cos(γ ± k )) 2 + r 2 0 sin 2 (γ ± k ) 2t ×   p ± i=0 sin(γ ± k ± i ) exp   - (r -r 0 cos(γ ± k ± i )) 2 + r 2 0 sin 2 (γ ± k ± i ) 2t     -1 , with U ∼ U([0, 1]).
Simulation in the case W does not exit the wedge: So far, we have provided a methodology in order to simulate the exit location and time for the simpler wedge V = ⟨α -, α + ⟩ with α + -α -= π/m. Now we discuss the case when W does not leave this wedge in the time interval [0, T ]. To simulate the final value W T of the Brownian motion starting from x 0 conditionally to the fact that it does not exit the wedge V on [0, T ], we need to simulate according to the density (10.3.3), which is a sum where some terms are negative. We propose the following method. 3. Simulate on which one of the two sets ∂V + n+1 or ∂V - n+1 , the process (W t ) t≥Tn starting from y n , at time T n , exits the wedge V n+1 and define θ n+1 := β ± n+1 according to the simulation result. This is done using a Bernoulli variable and the formula (10.5.5).

4. Simulate y n+1 ∈ ∂V n+1 , which is the value of W when it reaches ∂V n+1 for the first time after starting at y n at time T n , and knowing which one of the two events y n+1 ∈ ∂V ± n+1 has occurred, using formula (10.5.6). Denote the exit point as

y n+1 = (r n+1 cos(θ n+1 ), r n+1 sin(θ n+1 )).
5. Simulate τ n+1 , the time for the process W starting at y n = (r n cos(θ n ), r n sin(θ n )) at time T n to reach ∂V n+1 , knowing the event W Tn+τ n+1 = y n+1 . This can be done using the formula (10.5.7) and the acceptance-rejection procedure described after it. Define

T n+1 := T n + τ n+1 , so that W T n+1 = y n+1 .
If T n + τ n+1 < T then the algorithm iterates. This algorithm stops under one of these two conditions:

• Condition 1: T n + τ n+1 ≥ T . Then simulate the final value W T knowing that W Tn = y n and that for t ∈ [T n , T ], W t ∈ V n . We do this simulation conditionally to the event T n + τ n+1 > T and "forget" the exact simulated value of τ n+1 as well as the value of W Tn+τ n+1 . This is justified in Proposition 10.9.1, using

X := (W T ∧T n+1 , T n+1 1 T n+1 <T ) |Fn , Y := (W T n+1 , T n+1 ) |Fn and A = R 2 × [0, T ].
We simulate the value of W T using (10.5.9) and the acceptance-rejection procedure described after it.

• Condition 2: θ n+1 = 0 or θ n+1 = α. Then we obtain τ := T n+1 < T , i.e. the Brownian motion reaches the wedge D before the time T and the result of the simulation is

W T ∧τ = (r n+1 cos(θ n+1 ), r n+1 sin(θ n+1 )).
An illustration of this algorithm is given in Figure 10.4. Next, we prove that this algorithm can be carried out in a finite number of steps.

Proposition 10.5.2. Algorithm I ends in finite time, i.e. the number of required iterations to finish the algorithm, denoted by N , is such that N < ∞ almost surely. More precisely, for all Proof. The stopping times (τ i ) i follow the same law as ( τ i ), defined by τ 0 = 0 and: where m = π/Θ ∈ N, B = (B t ) t≥0 is a standard two-dimensional Brownian motion and θ(B t ) denotes the angle of the process2 B. Then:

K ∈ N, P x 0 (N ≥ K) ≤ 2 -⌊K/2⌋ . z n+1 y n+1 •yn β - n+1 β + n+1 α Θ/2 Θ/2
τ i+1 := inf t > 0 : |θ(B t+ τ i ) -θ(B τ i )| ≥ π 2m , ( 10
P x 0 ∞ i=1 τ i < ∞ = P x 0 ∞ i=1 τ i < ∞ .
But, if T := ∞ i=1 τ i < ∞, then there exist two random sequences, (t 1 n ) and (t 2 n ), increasing and converging to T , such that θ(B t i n ) = k i n π/(2m) + θ 0 for all n ∈ N and i ∈ {1, 2}, and where k i n ∈ Z has the same parity as i, so that the difference between θ(B t 1 n ) and θ(B t 1 n ′ ) is at least π/(2m) for all n, n ′ ∈ N. Taking n → ∞, we have necessarily that B T = 0, which occurs with probability 0 on any closed time interval.

Folding number in a wedge and complexity

In this section we study the complexity of Algorithms I and II in Sections 10.5.2 and 10.5.3 in separate cases. Recall that N denotes the number of iterations that an algorithm requires to finish one simulation, i.e. N := inf{n ∈ N : τ 1 + . . . + τ n > T }.

For Algorithm I, N is the number of times we touch the boundaries of the corresponding sets V n , n ∈ N, before touching one of the boundaries of D or reaching time T . For Algorithm II, N is the number of times that a Brownian motion reflected in a wedge V of angle π/(2m) goes from one boundary ∂V ± to another ∂V ∓ until time T . For this reason, we may call N the number of folds for the simulation of the reflected Brownian motion.

In this section, we give theoretical properties and bounds for N for each algorithm separately.

Majoration of N for the simulation algorithm of the stopped Brownian motion

For Algorithm I, by Proposition 10.5.2, we have for all K ∈ N, P x 0 (N ≥ K) ≤ 2 -⌊K/2⌋ , so

E x 0 [N ] = ∞ K=1 P x 0 (N ≥ K) ≤ ∞ K=1 2 -⌊ K 2 ⌋ = 3.
So Algorithm I ends in finite time almost surely and its complexity is finite in expectation.

Majoration of the number of folds of the Brownian motion in a wedge

We now investigate the complexity for Algorithm II. We decompose the standard two-dimensional Brownian motion W = (W t ) t≥0 in polar coordinates and introduce the notations that will be used in this section and the next one as follows. The successive stopping times (τ i ) i which appear in Algorithm II, have the same law as (τ i ) i , defined as in the proof of Proposition 10.5.3. Then we have N L = inf{n ∈ N : τ1 + . . . + τn > T }.

Moreover, using the skew-product representation of the Brownian motion (see [RY99, page 194]):

B t = R t U F (t) ,
(10.6.1)

where (R t ) t≥0 = (|B t |) t≥0 is a Bessel process of dimension 2, (U t ) t≥0 is a Brownian motion on S 1 ⊂ R 2 and F is defined as

F (t) = t 0 ds R 2 s , (10.6.2)
which is a strictly increasing process almost surely. Moreover, the processes (R t ) t≥0 and (U t ) t≥0 are independent. If we let (θ(U t )) t≥0 be the angle of the process (U t ) t≥0 then (θ(U t )) t≥0 is a standard one-dimensional Brownian motion. Define the stopping times (s i ) i by s 0 = 0 and

s i+1 = inf{t > 0 : |θ(U t+s i ) -θ(U s i )| ≥ π/(2m)}.
Then, the random variables s i are i.i.d. and have the law of the time for one-dimensional Brownian motion starting at 0 to reach the double barrier ±π/(2m). Then using (10.5.11) we have that for all K ∈ N,

K i=1 s i L = F K i=1 τi .
So the estimation of N can be simplified as

P x 0 (N ≥ K) = P x 0 K i=1 τi ≤ T = P x 0 K i=1 s i ≤ F (T ) . (10.6.3)
Note that F is a random change of time, independent of (s i ) i . Furthermore, the Laplace transform of s i is explicitly given in [JYC09, Section 3.5, Proposition 3.5.1.3] as

E[e -λs 1 ] = 2 e √ 2π √ λ 2m + e - √ 2π √ λ 2m .
In particular, we have:

(E[s i ], Var(s i )) = π 2 4m 2 , 2 3 π 2m 4 =: (µ, σ 2 ).
Proposition 10.6.1. Let 0 < a < 1 2 . Then we have

E x 0 [N a ] < ∞. On the other hand, if a ≥ 1/2 then E x 0 [N a ] = ∞.
Equation (10.6.3) together with the estimate in (10.9.3) give the result which is a consequence of the fact that the tails of F (T ) are large because the probability that the Bessel process R is close to zero is large and therefore due to (10.6.1) the simulation may spend a large time close to the origin.

Proof. We have that for all K ∈ N and δ > 1:

P x 0 (N a ≥ K) = µK 1/a δ 0 P x 0   ⌈K 1/a ⌉ i=1 s i ≤ y   P x 0 (F (T ) ∈ dy) + ∞ µK 1/a δ P x 0   ⌈K 1/a ⌉ i=1 s i ≤ y   P x 0 (F (T ) ∈ dy) =: I 1 (K) + I 2 (K).
For I 1 (K), we will use the fact that

2 exp π 2 4δm 2 < exp √ 2π 2m + exp - √ 2π 2m
for δ large enough. Then, we obtain that I 1 (K) decreases exponentially fast to zero. In fact, using Markov inequality

I 1 (K) ≤ P x 0   ⌈K 1/a ⌉ i=1 s i ≤ µK 1/a δ   ≤ e µK 1/a δ E x 0 e -s 1 ⌈K 1/a ⌉ ≤   2e π 2 4δm 2 e √ 2π 2m + e - √ 2π 2m   ⌈K 1/a ⌉ .
(10.6.4) Using (10.9.3), we have:

I 2 (K) ≤ P x 0 F (T ) ≥ µK 1/a δ ∼ K→∞ CK -1 2a , C = 1 √ 2Γ(1/2) ∞ r 2 0 2T e -u u du µ δ -1 2 .
(10.6.5) Since 0 < a < 1/2, the result follows. 10.6. Folding number in a wedge and complexity Similarly, in the case a ≥ 1/2, one has that

P x 0 (N a ≥ K) ≥ ∞ 2µK 1/a P x 0   ⌈K 1/a ⌉ i=1 s i ≤ y   P x 0 (F (T ) ∈ dy) .
For K large enough and y ≥ 2µK 1/a , we have P x 0 ( ⌈K 1/a ⌉ i=1 s i ≤ y) ≥ 1/2 by the Central Limit Theorem, and P x 0 F (T ) ≥ 2µK 1/a ∼ CK -1/(2a) as K → ∞ for some constant C > 0 which implies the result in this case.

From this result, we conclude that the average number of iterations of the algorithm in the reflected case is infinite. A modification of the above algorithm with finite number of iterations in expectation is provided below.

Proposition of modification of Algorithm II

When the simulated radii r 1 , . . . , r n in Algorithm II of Section 10.5.3 become small, the process (X t ) t∈[0,T ] goes from one boundary ∂V ± to the other boundary ∂V ∓ many times and the number of iterations becomes high. In Section 10.11 is hinted that this problem is in fact specific to the dimension 2. To remedy this problem, we study the asymptotic behavior of P x 0 (X t ∈ dy) when r 0 is close to zero. In fact, if the ratio rr 0 /t is small, then the density of the reflected Brownian motion expressed in (10.4.4) can be approximated by the distribution obtained by taking only n = 0:

P x 0 (X t ∈ dy) ≃ r tα e - r 2 +r 2 0 2t I 0 rr 0 t drdθ.
(10.6.6)

The function above can be renormalized so that it becomes a density which can be simulated using inequality (10.9.1) and the acceptance-rejection method with the reference density Then we modify Algorithm II as follows. We choose some small ε ∈ (0, r 0 ) and:

• Simulate the sequence of (r n ) n and (θ n ) n as in Algorithm II in Section 10.5.3.

• If after some iteration n, we have

r 2 n T -T n < ε (10.6.8)
(note that T -T n > 0 is the remaining time of the simulated process after step n), then we directly simulate the final point XT according to the approximation (10.6.6):

P x 0 ( XT ∈ dy| X Tn = y n ) = C(ε) r (T -T n )α e - r 2 +r 2 n 2(T -Tn) I 0 rr n T -T n drdθ, ( 10
.6.9)

C(ε) := α 0 ∞ 0 r (T -T n )α e - r 2 +r 2 n 2(T -Tn) I 0 rr n T -T n drdθ -1
.

Using that I 0 (x) ≥ 1 for all x ≥ 0, one obtains that C(ε) ≤ e r 2 n /(2(T -Tn)) . This upper bound for C(ε) is enough in order to implement the acceptance-rejection sampling method. In this case, the algorithm directly ends after one additional iteration. Proposition 10.6.2. Denote by N the number of iterations of the modified algorithm, (10.6.9), then for any 1 ≤ a < p < 2, there exists a quantity C(a, p, x 0 , T, m) > 0 such that for ε small enough:

E x 0 [ N a ] ≤ C(a, p, x 0 , T, m) ε p-1 .
(10.6.10)

Proof. We use a similar argument as in the proof of Proposition 10.6.1. Let ζ ε := inf {t ∈ [0, T ] : R 2 t /(T -t) ≤ ε} and let K ∈ N fixed. Using Markov's inequality, we have:

P x 0 ( N -1 ≥ K 1/a ) ≤ P x 0   ⌈K 1/a ⌉ i=1 s i ≤ F (T ∧ ζ ε ), F (T ∧ ζ ε ) ≤ µK 1/a δ   + P x 0 F (T ∧ ζ ε ) > µK 1/a δ ≤ e µK 1/a δ E x 0 e -s 1 ⌈K 1/a ⌉ + δ p (µK 1/a ) p E x 0 [F (T ∧ ζ ε ) p ] .
From here the result follows by using Lemma 10.9.5.

Proposition 10.6.3. Denote by XT , the result obtained from the above approximation algorithm. Then, for α ∈ (0, 2π) and ε small enough, the error made by the approximation in total variation distance satisfies:

d TV ( XT , X T ) ≤ C m ε min(1, π 2α ) ,
where C m is a constant which only depends on m.

Proof. If the approximation is not used, then XT = X T . Else, we denote by n the step when the approximation is used, so that we simulate the last step starting from (r n cos(θ n ), r n sin(θ n )) with remaining time T -T n =: t ′ . Since the ε-condition is reached, we have r 2 n /t ′ < ε. Then if we denote by d TVn ( XT , X T ) the total variation distance conditioned to the filtration F n up to time T n , and d n := 1/2 for n = 0 and d n := 1 otherwise, we have: Using that for all x ≥ 0, I 0 (x) ≥ 1, we have C(ε) ≤ e r 2 n /(2t ′ ) . On the other hand, we have

d TVn ( XT , X T ) ≤ D (C(ε) -1 + 1) r t ′ α e -r
C(ε) -1 ≤ 1 t ′ ∞ 0 re -r 2 /(2t ′ ) I 0 rr n t ′ dr = ∞ 0 ue -u 2 /2 I 0 r n u √ t ′ du.
Using (10.2.1) one immediately obtains that |I ′ 0 (x)| + |I ′′ 0 (x)| ≤ 2e x . Therefore the function A(δ) := ∞ 0 ue -u 2 /2 I 0 (δu)du, δ ≥ 0 is twice differentiable. As I 0 (0) = 1 and I ′ 0 (0) = 0, we obtain A(0) = 1, A ′ (0) = 0 and then A(δ) = 1 + O(δ 2 ) as δ → 0 which gives C(ε) ≥ 1 -Cε for small enough ε > 0. These bounds on C(ε) imply that |C(ε) -1|/(C(ε)ε) is bounded. For the second term, using the expression (10.2.1) and a multiplicity property of the number of terms of the type kπ/α in the interval [0, k], we have u β e -(γ-1)u 2 du + 1 2(γ -1)

β 0 -1 2(γ-1)
e -(γ-1)u 2 du ≤ 4αγ π e 1 4(γ-1)

1 √ γ -1 1+β ∞ 0 v β e -v 2 dv + 1 2 β (γ -1) β+1/2 0 -1 2 √ γ-1 e -v 2 dv .
The first term converges to zero as O(ε π/(2α) ) while the second term converges to zero as O(ε 1+π/α ) when ε → 0. Taking all the elements into account we obtain that d TV ( XT , X T ) =

E x 0 e -b• W T ∧ τ + b 2 T ∧ τ 2 f ( W T ∧ τ , T ∧ τ ) = E x 0 [f (W T ∧τ , T ∧ τ )] .
So we have:

E x 0 f ( W T ∧ τ , T ∧ τ ) = E x 0 e b•W T ∧τ -b 2 T ∧τ 2 f (W T ∧τ , T ∧ τ ) .
To simulate the Brownian motion with drift stopped in the wedge D, we proceed as follows:

1. Simulate W T ∧τ and T ∧ τ for the stopped process without drift according to Algorithm I in Section 10.5.2. We can also deduce an explicit formula for the density of the stopped Brownian motion with drift using (10.3.4). In fact, for y ∈ D, we have (10.7.3) with the same notations as before: τ := inf{s : W s + bs ∈ ∂D}, x = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) and y = (r cos(θ), r sin(θ)).

The result of the simulation becomes

Adaptation of the simulation algorithms for Itō processes

In this section, we present some extensions of the above simulation methods applied to the approximation of stopped and reflected diffusions at some fixed time T > 0. Let us start, first with the case of killed diffusion.

Consider the following Itō process in R 2 : (10.7.4) where (W t ) t≥0 is a standard two-dimensional Brownian motion and x 0 ∈ D. We define τ to be the exit time of (Y t ) t≥0 from D.

dY t = b(Y t )dt + σ(Y t )dW t Y 0 = x 0 ,
Simulation of Y T ∧τ : Choose t i := T i/n, i = 0, ..., n, a uniform partition of [0, T ] and consider the following Euler-Maruyama scheme. Given Ȳt k ∈ int(D) (with Ȳt 0 = x 0 ) and supposing that τ := inf{s; Ȳs ∈ ∂D} > t k , simulate Ȳt k+1 as the final value of the following Brownian motion with drift, stopped on the wedge D, t ∈ [t k , t k+1 ]: Note that we have to take into account the decorrelation step and the change of angle as described in Section 10.3 at every step. If the process (10.7.5) exits the wedge D in the interval [t k , t k+1 ] then the algorithm directly stops. This scheme has weak order one. That is, for any f, b, σ ∈ C 5 b ( D) one has that there exists a constant C f > 0

E x 0 [f (Y T ∧τ )] -E x 0 f ( ȲT ∧τ ) ≤ C f n -1 .
The proof is straightforward if one follows the same line of proof as in [START_REF] Gobet | Euler schemes and half-space approximation for the simulation of diffusion in a domain[END_REF]. In particular, see Section 2.2.1 and note that our case is simpler as the proposed scheme does not have the possibility of touching the boundary before it is stopped or reaches T . For a discussion about the associated partial differential equation with Dirichlet conditions, see the discussion right after (10.10.1).

Since the Girsanov change of measure uses an exponential function, it may not be suitable to processes with large drifts as it may increase the variance. For this reason, we also propose a two-step Euler-Maruyama scheme for the stopped process.

Simulation of Y T ∧τ : Choose t i := T i/n, i = 0, ..., n, a uniform partition of [0, T ] and consider the following two-step Euler-Maruyama scheme. Given Ȳt k ∈ int(D) (with Ȳt 0 = x 0 ) and supposing that τ := inf{s; Ȳs ∈ ∂D} > t k and that τ := inf{s; Ỹs ∈ ∂D} > t k , simulate Ȳt k+1 ∧τ in two steps as follows for t ∈ [t k , t k+1 ]:

Ỹt∧τ := Ȳt k + b( Ȳt k )(t ∧ τ -t k )
(10.7.7) Ȳt∧τ := Ỹt k+1 ∧τ + (W t∧τ -W t k ).

(10.7.8)

The second step is performed using the algorithm in Section 10.5.2. Note that we have to take into account the decorrelation step and the change of angle as described in Section 10.3. If the process exits the wedge D in the interval [t k , t k+1 ] then the algorithm directly stops. This scheme has weak order one. That is, for any f, b, σ ∈ C 5 b ( D), there exists a constant C f > 0 such that

E x 0 [f (Y T ∧τ )] -E x 0 f ( ȲT ∧τ ) ≤ C f n -1 .
(10.7.9)

The proof is similar to the proof for the convergence of the two-step Euler scheme for the reflected processes in Proposition 10.7.1. Now let (Z t ) t≥0 to be the solution of a stochastic differential equation with drift coefficient b which is normally reflected on the boundary of the wedge D. That is,

dZ t = b(Z t )dt + dW t + dL t .
For details on existence and uniqueness for this equation we refer to [START_REF] Saisho | Stochastic differential equations for multidimensional domain with reflecting boundary[END_REF].

In this case, we do not know how to extend the simulation method using the Girsanov change of measure; instead we propose a two-step scheme.

Simulation of Z T : Choose t i := T i/n, i = 0, ..., n a uniform partition of [0, T ] and consider the following two-step scheme. Given Zt k (with Zt 0 = x 0 ), simulate Zt k+1 as the final value of the reflection on the wedge D of the Brownian motion and drift in two steps as follows for t ∈ [0, t k+1 -t k ] The terms L i , i = 1, 2, denote the respective local time terms for each of the two steps. The simulation of the second step of this algorithm uses the argument described in either Sections 10.5.3 or 10.6.3. The weak rate of convergence for these methods is as follows:

Zt+t k := Zt k + b( Zt k )(t -t k ) + L 1 t -
Proposition 10.7.1. Assume that f, b ∈ C 5 b ( D) then

E x 0 [f (Z T )] -E x 0 f ( ZT ) ≤ C f n -1 .
(10.7.12)

Denoting by ẐT , the result of the algorithm using the approximation described in Section 10.6.3, we have

E x 0 [f (Z T )] -E x 0 f ( ẐT ) ≤ C f n -1 + ε min(1, π 2α ) .
(10.7.13)

Proof. The proof of (10.7.13) is a small modification of (10.7.12). Therefore, we prove the rate of convergence for the algorithm provided in Section 10.5.3. The argument follows as in the classical diffusion case which can be found in the proof of the main result in [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF]. We use this argument and refer the reader to [START_REF] Talay | Expansion of the global error for numerical schemes solving stochastic differential equations[END_REF] for more details.

First, consider u to be the solution of the following backward partial differential equation with Neumann conditions: ∂ t u(t, x) + Lu(t, x) = 0, (t, x) ∈ [0, T ] × D u(T, x) = f (x),

x ∈ D ∇u(t, x) • n(x) = 0, t > 0, x ∈ ∂D.

(10.7.14)

Here, Lu(t, x) := b(x)∇u(t, x) + 1 2 ∆u(t, x). Following the same discussion as in (10.10.1), one obtains that u ∈ C 7/2,7 ([0, T ] × D). This property will be used when doing Taylor's expansions for u. Next, note that

E x 0 [f (Z T )] -E x 0 f ( ZT ) = n-1 i=0 E x 0 u(t i , Zt i ) -u(t i+1 , Zt i+1 ) =: - n-1 i=0 A i .
(10.7.15)

We will now consider each term within the above sum. Using Itō-Tanaka formula (here one uses (10.7.14) to cancel all local time terms), we have Each of the above derivatives of u is bounded. In the case of the increments of Zs -Zt i = (W s -W t i ) + L 2 s -L 2 t i + Zt i+1 -Zt i one has to take one further step in the Taylor expansion to be able to use the fact that the expectation of the Brownian increment is zero. Namely,

A i = E x 0 u(t i ,
1 0 ∇∂ t u(t i , α Zs + (1 -α) Zt i ) • ( Zs -Zt i )dα = ∇∂ t u(t i , Zt i ) • ( Zs -Zt i ) + 1 0 (1 -α)( Zs -Zt i ) ⊤ • ∇ 2 ∂ t u(t i , α Zs + (1 -α) Zt i ) • ( Zs -Zt i )dα.
After this and canceling the first terms of the expansion using (10.7.14) and taking expectations we see that the sum of most terms is of order O(n -2 ). The remaining terms are bounded by C t i+1 t i (L j s -L j t i )ds for j = 1, 2. Considering these terms within (10.7.15), one obtains for L j T = n-1 i=0 (L j t i+1 -L j t i ) that these sums are bounded by

n-1 i=0 t i+1 t i E x 0 [L j s -L j t i ]ds ≤ n -1 E x 0 [L j T ] ≤ Cn -1 ,
where we used that t → L j t is increasing. The last inequality follows from an estimate for the Skorohod problem in convex domains (see [START_REF] Saisho | Stochastic differential equations for multidimensional domain with reflecting boundary[END_REF]Theorem 4.2]). In the case of the algorithm described in Section 10.6.3 one may modify the above arguments adding at each time t i , the possibility of ending the simulation in the partition interval using the criteria provided in (10.6.8). Using Proposition 10.6.3 one obtains the result. 10.2: Simulations with α = 0.58, r 0 = 3, θ 0 = 0.4, f (r cos(θ), r sin(θ)) = sin 2 (θ), T = 1.

Simulations

We implement the algorithms3 described in Sections 10.5.2 and 10.5.3 as well as the approximated version in Section 10.6.3. We also give simulation results obtained with Metzler's algorithm [Met09, Section 2.1]. This last algorithm simulates W τ (instead of W τ ∧T ) directly for a wedge of general angle α, by: 1. Simulate the radius |W τ | using (10.5.6).

2. Conditionally simulate τ by acceptance-rejection, approximating the infinite sum by a partial sum, and using a Cauchy distribution as reference density.

However, Metzler's algorithm cannot be adapted to simulate W τ ∧T and is biased for the simulation of τ (see [Met09, Proposition 2.1.8]). In Tables 10.1 and 10.2, we consider the Monte Carlo estimation of various expectations. We choose positive test functions in order to avoid cancellations and r 0 so that E x 0 [τ ] is of the same order as T , so that the stopped and reflected processes are significantly different than the standard Brownian motion.

We note that for the estimation of E[f (W τ )], our simulation method finds a similar value as the algorithm proposed by Metzler, as both are exact simulation methods. However, for the estimation of E[τ ], the bias in the method proposed by Metzler seems to be significant.

As a second way to check our algorithm, we also simulated the projection on the first axis of W τ ∧T . Note that the simulation gives a result close to the initial point 1.5 • cos(0.3) ≃ 1.4330 which is reported on the sixth line of Table 10.1. This is because t → W τ ∧t is a martingale.

For the estimation of E[f (X T )] (reflected Brownian motion), the exact algorithm takes too much time, as hinted in Proposition 10.6.1, hence the need to use the approximation version from Section 10.6.3. We only give a simulation example with 1000 samples, as we could not get a more proper estimation. Indeed, if we increase the number of samples, there appear trajectories where N becomes large and where the algorithm does not stop in reasonable time. , with α = 0.9, r 0 = 1.5, θ 0 = 0.3, µ = (0.1, 0.2), κ = (0.7, 0.5), σ = I 2 , f (x, y) = x 2 + y 2 , T = 1. Z corresponds to the reflected process in the same domain. The first stopped process is simulated using the Girsanov method while the second one using the two-step Euler-Maruyama scheme.

To have a better idea of the behavior of the algorithm, we provide on the left side of Figure 10.6, the histogram of N for the exact Algorithm II from Section 10.5.3; if, for a trajectory, N exceeds 150, then we stop the algorithm. We observe that for most of the trajectories, N is not too large (does not exceed 5). However, for a few iterations (around 0.2%) which corresponds to the case where the radius r becomes small, N is large, which slows down the algorithm. On the right side of Figure 10.6 we trace E x 0 [N ] for the approximated reflected algorithm in function of ε. However, we could not trace the bias in function of ε, as it seems negligeable in comparison with the size of the confidence interval. This is why we do not need to lengthen the simulation time of each trajectory by choosing a very small ε; ε = 0.02 or ε = 0.05 with E x 0 [N ] ≈ 5 is sufficient.

In Table 10.3, we implement the algorithms adapted to general Itō processes from Section 10.7.2.

Appendix: Auxiliary Lemmas

We start with a simple lemma about the simulation of a r.v. X using only partial information of a previously simulated r.v. Y. Proposition 10.9.1. Let X and Y be two random variables. We want to simulate X. Let A be a deterministic set. We consider the following algorithm :

1. Simulate Y.

2. If Y ∈ A, simulate X conditionally to the value of Y in the previous step, and return X. 10.9. Appendix: Auxiliary Lemmas

If Y /

∈ A, simulate X conditionally to the event Y / ∈ A, and return X.

Let us denote X the value returned by the algorithm. Then:

X L = X.
Proof. The random variable X is defined conditionally to Y as:

Estimates on Bessel processes

Proposition 10.9.2. For all x ≥ 0, we have:

1 ≤ I 0 (x) ≤ e x .
(10.9.1) Furthermore, for x ≥ 0 and ν ≥ 0, we have: (10.9.4)

I ν (
Step 2: In fact, we prove the asymptotic expansion:

E r 0 e -λ 2 2 F (T ) = λ→0 + 1 - λ 2 ∞ r 2 0 2T
e -u u du + O(λ 2 ). (10.9.5)

We have for all x > 0, Γ(x + 1) = xΓ(x) and Γ(1) = 1 so that Γ(x)

∼ x→0 + 1 x .
Then, using (10.9.4) and the change of variable w = T u/r 2 0 :

E r 0 e -λ 2 2 F (T ) = 1 Γ λ 2 r 2 0 2T 0 u λ 2 -1 1 - 2T u r 2 0 λ/2
e -u du.

Using the definition of the Gamma function and decomposing the integral in two parts, we have

1-E r 0 e -λ 2 2 F (T ) = 1 Γ λ 2   r 2 0 2T 0 u λ 2 -1 e -u 1-1- 2T u r 2 0 λ/2 du + ∞ r 2 0 2T u λ 2 -1 e -u du   = : I 1 +I 2 .
We use the expansion (1 -x) α = ∞ k=0 (α log(1 -x)) k /k! for |x| < 1 and α → 0 in order to analyze the first integral. This gives the following asymptotic equivalence

I 1 = λ→0 + - λ 2 4 r 2 0 2T 0 u -1 e -u log 1 - 2T u r 2 0 du + o(λ 2 ) = O(λ 2 ).
And on the other side:

I 2 = 1 Γ λ 2 ∞ r 2 0 2T u λ 2 -1 e -u du = λ→0 + λ 2 ∞ r 2 0 2T
e -u u du + O(λ 2 ).

From here we obtain the asymptotic expansion (10.9.5).

Step 3: We get (10.9.3) from the expansion (10.9.5) and using Karamata Tauberian theorems (see [START_REF] Embrechts | Modelling extremal events[END_REF], Corollary A3.10): Proposition 10.9.4. Let df be a measure on R + , F (x) := x 0 df , f (λ) := ∞ 0 e -λx F (x)dx, 0 ≤ γ < 1 and let L > 0. Then the following are equivalent:

1. 1 -f (λ) ∼ Lλ γ , λ → 0 + 2. 1 -F (x) ∼ L Γ(1 -γ) x -γ , x → ∞.
Next we will give an estimate for E[F (T ∧ ζ ε ) p ] in the case of the modified algorithm for reflected Brownian motion. Lemma 10.9.5. We define the stopping time ζ ε := inf{t ∈ [0, T ] : R 2 t /(T -t) ≤ ε}. Then we have the following bound for p ∈ (1, 2) and ε > 0 small enough: where ν = d 2 -1 and d is the dimension of the Bessel process. Taking d = 2, we have: Then, using (10.9.1) for both I 1 and I 2 : On the other hand, u and ũ satisfy the same boundary equation on D. Indeed, for all x ∈ D, f (x) = f (x) so u and ũ satisfy the same initial conditions. For x ∈ ∂D -, k = 0, . . . , 2m -1 and y ∈ D, we have Proof of Theorem 10.4.2 in the general case. As in the above proof we consider u(t, x) = E x [f (X t )] , (10.10.2)

E r 0 [F (T ∧ ζ ε ) p ] ≤ C 1 + log 2T r 2 0 (εT )
E r 0 [F (T ∧ ζ ε ) p ] ≤ T 0 E r 0 1 R 2p t 1 Rt≥ √ ε(T -t) dt = T 0 ∞ √ ε(T -t) 1 r 2p-1 t e -
I 1 ≤ ∞ √ εT 1 r 2p-1 T 0 1 t e -(
where t ≥ 0, x ∈ D, and f : D → [0, +∞) is a non-negative continuous function with compact support. Then u is the solution of the partial differential equation (10.10.1). Considering the formula obtained for the case α = π/m above, we would like to express it so that it does not depend explicitly on m. We first assume that α = π/m for some m ∈ N. In order to switch to polar coordinates, let x = (r 0 cos(θ 0 ), r 0 sin(θ 0 )), y = (r cos(θ), r sin(θ)) and recall that ϑ k for 0 ≤ k ≤ 2m -1 is the angle of T k (y) in [0, 2π), i.e. T k (y) = (r cos(ϑ k ), r sin(ϑ k )) with ϑ 2k = 2kα + θ and ϑ 2k+1 = 2(k + 1)α -θ. We then rewrite (10.4.3): which gives a formula which does not depend on m. Now, we have to check that this formula is well defined for any α ∈ (0, π) and that it is the unique solution the partial differential equation (10.10.1) with D = ⟨α⟩. This is explained in Corollary 10.10.1. In our current situation, d = 2, the eigenvalues are λ j := (jπ/α) 2 for j ≥ 0 and the eigenfunctions are m j (θ) = 2/α cos (jπθ/α), j ≥ 1 and m 0 ≡ 1/ √ α.

P x (X t ∈ dy) = r 2πt e -
Corollary 10.10.1. Consider a general d-dimensional cone generated by all rays emanating from the origin and passing through a compact subset D ⊂ S d-1 which has smooth boundary. Consider X to be the normally reflected Brownian motion at the boundary of the cone. Then X t has a density given by To explain the elements in the above formula, denote by L S d-1 , the Laplace-Beltrami operator on S d-1 . With the above assumptions, there exists a complete set of orthonormal eigenfunctions m j with corresponding eigenvalues 0 ≤ λ 0 < λ 1 ≤ λ 2 < . . . satisfying L S d-1 m j (x) = -λ j m j (x) for x ∈ D ∇m j (x) • n(x) = 0 for x ∈ ∂D,

P x (X t ∈ dy) = re -
α j = λ j + d 2 -1 2 .
Proof. The beginning of the proof is the same as in the statements following (10.10.1) in what refers to the existence and uniqueness of the associated PDE. Existence and uniqueness for the reflected process in the generalized cone can be deduced from [START_REF] Bass | Uniqueness for the Skorokhod equation with normal reflection in Lipschitz domains[END_REF] (see also [START_REF] Bass | Uniqueness for reflecting Brownian motion in lip domains[END_REF]Remark 4.1] and the references therein).

In order to prove that (10.10.5) satisfies the associated PDE, one follows a similar proof in [START_REF] Bañuelos | Brownian motion in cones[END_REF] for the killed case. The proof in the reflected case follows line by line, the proof in [START_REF] Bañuelos | Brownian motion in cones[END_REF], except that the required estimates and properties of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator in the case of Neumann boundary conditions have to be referred to the proper literature (for this see, [START_REF] Grebenkov | Geometrical structure of Laplacian eigenfunctions[END_REF], [START_REF] Grieser | Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary[END_REF] and [START_REF] Kroger | Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space[END_REF]) although the estimates do not change as far as it relates to the proof of [BnS97, Lemma 1] with the corresponding corrections of typos. In order for the required estimates to be satisfied one needs that the generalized d-dimensional cone is generated by all rays emanating from the origin and passing through a compact subset D ⊂ S d-1 which has smooth boundary.

We also remark that the density expressions in [START_REF] Bañuelos | Brownian motion in cones[END_REF] are written under polar measures which explains why our expressions have an extra r which appears due to the Jacobian of the change of coordinates.

We also provide a full elementary proof that (10.10.4) satisfies the initial conditions in the Supplementary Material.

Appendix: A hint for higher dimensions

The following proposition is provided so as to hint at the possibilities in dimension higher than two. It shows that dimension two is the case which is mathematically difficult to treat. Proof. The density of R t in r is given in (10.9.7), so by performing the change of variable r = ut, we have Note that for all ε > 0, where for we used the Proposition 10.9.2 for the convergence of the second integral. So to prove the convergence of the integral, we only need to prove the convergence for the integral in t around zero. To do so we use Proposition 10.9.2 again so that for ν, r 0 > 0:

E
E r 0 [F (ε)] ≤ 1 r ν 0 ε 0 t ν-1 e - r 2 0 2t
∞ 0 e ur 0 + 1 π(ν + ur 0 ) e -tu 2 2 u ν-1 dudt < ∞.

In fact, using that ν > 0, we have

I 1 := 1 r ν 0 ε 0 t ν-1 ∞ 0 e -t 2 (u-r 0 t ) 2 u ν-1 dudt ≤ 1 r ν 0 ε 0 t ν-1 ∞ -∞ e -t 2 u 2 |u| + r 0 t ν-1 dudt = 2 r ν 0 ε 0 t ν-3/2 ∞ 0 e -u 2 /2 u √ t + r 0 t ν-1 dudt ≤ 2 r ν 0 ε 0 t ν-3/2 ∞ 0 e -u 2 /2 u + r 0 t ν-1 dudt = 2 r ν 0 ε 0 t -1/2
∞ 0 e -u 2 /2 (u + r 0 ) ν-1 dudt < ∞.

As for the second integral, we have

I 2 := 1 r ν 0 ε 0 t ν-1 ∞ 0 1 π(ν + ur 0 ) e -tu 2 2 - r 2 0 2t u ν-1 dudt ≤ 1 r ν 0 ε 0 t ν-1 e - r 2 0 2t dt ∞ 0 u ν-1 π(ν + ur 0 ) du < ∞.
10.12 Supplementary material 10.12.1 Proof of the convergence to the initial condition when t → 0

In this section, we prove the following result:

Proposition 10.12.1. The formula (10.4.4) satisfies the initial conditions of the heat equation, i.e. for all functions f : D → R continuous with compact support, r 0 > 0 and θ 0 ∈ (0, α): Let us denote by A 1 and A 2 the two terms appearing in the integral representation in (10.12.2) after replacing with the above formula, with d n = 1 2 if n = 0 and d n = 1 for n ≥ 1: Proposition 10.12.2. We have A 2 -→ t→0 0.

A 1 := 2 απ D r t f (r, θ)e -
Proof.

Step 1: For the moment let us assume that the integration order can be switched. For all a ∈ R and b > 0, we have: Now we prove that for all j, k ∈ {0, 1}, we have: We will do the analysis of J j,k dividing the integration region in two: J 1 j,k which comprises the integral on the region D × (1, ∞) and the remaining which is denoted by J 2 j,k . With this in mind, note that g is continuous on [0, ∞) × (0, α) and is locally integrable in (0, 0). In fact, for all a ∈ R and b > 0: On the other hand, using (10.12.5) and the continuity of g we obtain Step 2: Now, we prove that one can interchange the order of the integrals and sum in A 2 . Note that Fubini's theorem does not apply here because of the factors cos(nπθ/α) and cos(nπθ 0 /α). A computation similar to (10.12.3) leads to, for all N ∈ N: (1 -e -b cos(a)) 2 + e -2b sin 2 (a) .

J j,k := 1 t D r f (r, θ)e -
|J 2 jk | ≤ ||f || ∞ 1 t ∞ 0 re -
Next, we prove that, for all j, k ∈ {0, 1} and for all t > 0, denoting R N jk the difference between the infinite sum and the partial sum up in A 2 to N , ) is integrable in (0, 0). We denote R N,1 jk and R N,2 jk the two terms obtained after splitting the integral with respect to u on (0, 1) and (1, ∞) respectively. We have then From the above arguments we obtain the conclusion: R N jk → 0 for all j, k ∈ {0, 1}. Proposition 10.12.4. We have A 1 -→ t→0 f (r 0 , θ 0 ).

R N jk := 1 t D re -
|R N,2 jk | ≤ 1 t ||f || ∞ ∞ 0 re -
Proof. First, let us assume that the integration order can be exchanged. This will be further discussed in Step 5.

Step 1 : We exchange the order of integrals in A 1 as: We will study the limit of A 1 in the above integral order. First, we treat the sum inside f 2 (u, t) using trigonometric identities for cos nπ(θ 0 ±u) α . We see that is enough to find the limit for: In our case, we will use the following normalization of the classical Fourier inversion formula:

A 1 = π 0 f 1 (u, t)f 2 (u, t)du,
Theorem 10.12.5 (Fourier Formula). Let g : R → R be a periodic and continuous function of period 2L. Then the following series converges for all x ∈ R and: Next, we extend the definition of the function f . For all r ≥ 0 and θ ∈ [0, α], we define f (r, -θ) := f (r, θ), and we make it 2α-periodic by defining f (r, θ + 2kα) = f (r, θ). This way, f is an even and 2α-periodic function and is still continuous. With this definition, and using the Fourier inversion formula, we get for θ 0 ± u ∈ (-α, α): (10.12.6)

Step 2 : Next, let us study f 2 (u, t) in two separate cases the integral in r: And the second term is, up to the multiplicative constant √ 2π, an approximation of the unity around r = r 0 cos(u) > 0, so that in this case, the integral in r converges to √ 2πr 0 cos(u) f (r 0 cos(u), θ 0 ± u).

Case 2: If r 0 cos(u) ≤ 0: The total mass of 1 √ t re -(r-r 0 cos(u)) 2 /(2t) is bounded above by 1

√ t ∞ 0 re -r 2 2t dr = √ t -→ t→0 0,
so that the integral in r converges to 0. We remark here that convergences in the above two cases are uniform with respect to u within their respective domains.

Step 3 : From the previous step, we consider now the integral with respect to u. Taking into consideration the previous step, we can restrict to the case cos(u) > 0 or equivalently u ∈ [0, π/2]. That is, consider Thus, up to the multiplicative constant π/(2r 2 0 ), u → e -r 2 0 sin 2 (u)/(2t) cos(u) is an approximation of the unity around u = 0, so that

I ± := 1 √ t π/2 0 f (r 0 cos(u), θ 0 ± u)e -
I + + I --→ t→0 2π r 2 0 f (r 0 , θ 0 ).
Step 4 : Now, we put all previous steps together. In Step 1, we proved that for all t > 0: We have proved in Step 2 that for all u, f 2 (u, t) converges when t → 0 to We have:

π 0 f 1 (u, t)f 2 (u, t)du -f 1 (u, t)f 2 (u)du ≤ sup t π 0 f 1 (u, t)du ||f 2 (•, t) -f 2 (•)|| ∞ ,
and ||f 2 (•, t) -f 2 (•)|| ∞ → 0 since in Step 2, the convergence is uniform with respect to u.

Step 5 : We now prove that for all r 0 , θ 0 , t fixed, the integration order can be switched. Note that for fixed n, by Fubini's theorem, the integration order in r, θ and u can be switched, as So that the order of integration can be exchanged for every partial sum. Now, for N ∈ N: Although the extension of f on [-α, α] is not an element of C 1 , we can perform the integration by parts on each interval [0, α] and [-α, 0]. Then, || fN (r, •) -f (r, •)|| 2 → 0 as N → ∞ uniformly in r. That way we obtain the convergence to 0 of the difference between the partial sum and the series.

  10.6.1 Majoration of N for the simulation algorithm of the stopped Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.2 Majoration of the number of folds of the Brownian motion in a wedge . . 10.6.3 Proposition of modification of Algorithm II . . . . . . . . . . . . . . . . . 10.7 Adaptation of the algorithms to general processes . . . . . . . . . . . . . . . . . . 10.7.1 Stopped Brownian motion with constant drift . . . . . . . . . . . . . . . . 10.7.2 Adaptation of the simulation algorithms for Itō processes . . . . . . . . . 10.8 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Appendix: Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9.1 Estimates on Bessel processes . . . . . . . . . . . . . . . . . . . . . . . . . 10.10Appendix: Proofs in Section 10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.11Appendix: A hint for higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 10.12Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.12.1 Proof of the convergence to the initial condition when t → 0 . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Gradient descent algorithm.

Figure 1 Figure 1

 11 Figure 1.3: Scheme of a fully connected neural network with 3 layers (2 hidden layers) and with output in R 2 .

  φ(Z + θ)e -|θ| 2 2 -⟨θ,Z⟩ = E φ 2 (Z + θ)e -|θ| 2 -2⟨θ,Z⟩ -E φ(Z + θ)e -|θ| 2 2 -⟨θ,Z⟩ 2 = E φ 2 (Z + θ)e -|θ| 2 -2⟨θ,Z⟩ -E[φ(Z)] 2 .The objective is to solve the minimization problem Minimize θ∈R d

Figure 1

 1 Figure1.6: Example of optimal quantization in R 2 where µ is the unweighted Gaussian mixture with component distributions N (y k , (1/4)I 2 ), 1 ≤ k ≤ 5, y k = (cos(2kπ/5), sin(2kπ/5)) and where the number of quantizers is K = 5. We plot the density of µ, the points y k in blue, the centroids x k given by the CLVQ algorithm in red and the frontiers of the Voronoï partition (V k ) 1≤k≤5 in red.

  R d |x -y| p π(dx, dy) 1/p : π ∈ P([ XN ], [X T ]) , where P([ XN ], [X T ]) stands for the set of probability distributions on (R d × R d , B(R d ) ⊗2 ) with respective marginal distributions [ XN ] and [X T ], or as sup

Figure 1 . 7 :

 17 Figure 1.7: Example of trap for the gradient descent.

  is C 1 and uniformly elliptic i.e.∃ σ0 > 0, ∀x ∈ R d , σ(x)σ ⊤ (x) ≥ σ0 ,and bounded with bounded partial derivatives. If the drift b satisfies b(x) = -(σσ ⊤ ∇V )(x) + a 2

  These results have been presented at the Machine Learning conference International Joint Conference on Neural Networks IJCNN 2023 and are published in the conference proceedings as a joint work with Gilles Pagès and under the title Langevin algorithms for Markovian Neural Networks and Deep Stochastic Control.

Figure 2 . 3 :

 23 Figure 2.3: Performance of preconditioned Langevin algorithms compared with vanilla SGLD on the MNIST dataset. The values of the hyperparameters are a(n) = A log -1/2 (c 1 n + e) with A = 2.10 -3 and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n) where c 2 n = 1 after 5 epochs and where for SGLD, γ 1 = 0.001 for L-RMSprop and L-Adam and γ 1 = 0.1 for L-Adadelta.

Figure 2 . 4 :Figure 2 . 5 :

 2425 Figure 2.4: Side-by-side comparison of optimization algorithms with their respective Langevin counterparts for the training of a deep neural network on the MNIST dataset. We display the performance of SGD for reference. The values of the hyperparameters are a(n) = A log -1/2 (c 1 n+ e) with A = 1.10 -3 for L-Adam and A = 5.10 -4 for L-RMSprop and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n) where c 2 n = 1 after 5 epochs and where γ 1 = 0.01 for SGLD and γ 1 = 0.001 for the others.

  Figure 3.1: Comparison of Langevin optimizers for the training of ResNet20 on the CIFAR-10 dataset. Batch size is 512. The schedules are γ n = γ 1 /(1 + c 0 n) where γ 1 = 5 e -3 and c 0 n = 1 after n = 20 epochs, and a n = A log -1/2 (c 0 n + e) where A = 5 e -3.

  For a, b ∈ R d , we denote by a * b the element-wise product, i.e. ∀i ∈ {1, . . . , d}, (a * b) i = a i b i .

  Theorem 4.3.2[START_REF] Athreya | Gibbs measures asymptotics[END_REF]. Let f : R d → R be a Borel function and let x ⋆ ∈ R d such that :

  Table 4.2: Illustration of the subspaces4.4.7 Proof of Theorem 4.4.1 for p = 4 Theorem 4.4.5. Let

  -example and proof of Theorem 4.4.1 with p ≥ 5 under the hypothesis (4.4.6)

4. 7 .Figure 4 . 1 :

 741 Figure 4.1: Distribution of the eigenvalues of the Hessian matrix at the end of training.

  def compute_hessian ( model , inputs , targets ) : with tf . GradientTape () as g1 : with tf . GradientTape () as g2 : loss_value = model . loss ( inputs , targets ) grads = g2 . gradient ( loss_value , model . tr a i na b l e_ v a ri a b le s ) hessian_matrix = g1 . jacobian ( grads , model . t r ai n a bl e _ va r i ab l e s ) return hessian_matrix

class

  CustomLayer ( tf . keras . layers . Layer ) : def __init__ ( self , x_star , direction_vectors , activation = None ) : super ( CustomLayer , self ) . __init__ () self . x_star = x_star # x_star [0] is the kernel matrix and x_star [1] is the bias self . di rectio n_vect ors = tf . reshape ( direction_vectors , [ dir ection _vecto rs . shape [0] , x_star [0]. shape [0] , x_star [0]. shape [1]]) self . activation = activation def build ( self , input_shape ) : self . kernel = self . add_weight ( " kernel " , shape = [ direct ion_ve ctors . shape [0] ,]) def call ( self , inputs ) : outputs = tf . matmul ( inputs , self . x_star [0] + tf . tensordot ( self . kernel , self . direction_vectors , axes =[[0] ,[0]]) ) + self . x_star [1] if self . activation is not None : outputs = self . activation ( outputs ) return outputs

Figure 5 . 1 :

 51 Figure 5.1: The Sigmoid and ReLU functions

  . keras . losses . S p a r s e C a t e g o r i c a l C r o s s e n t r o p y ( from_logits = True ) )

  def _ r e s o u r c e _ a p p l y _ d e n s e ( self , grad , var ) : var . assign ( var -self . learning_rate * grad )

)

  or self . _ f a l l b a c k _ a p p l y _ s t a t e ( var_device , var_dtype ) lr_t , sigma = coefficients [ " lr_t " ] , coefficients [ " sigma " ] return coefficients , lr_t , sigma Then the L-RMSprop algorithm is given as a subclass of LangevinOptimizer by: class LRMSprop ( Lange vinOp timize r ) : def __init__ ( self , learning_rate =0.001 , sigma =0.001 , alpha =0.9 , diagonal_bias =1 e -6 , name = " LRMSprop " , ** kwargs ) : super () . __init__ ( learning_rate , sigma , name , ** kwargs ) self . alpha = alpha self . diagonal_bias = diagonal_bias def _create_slots ( self , var_list ) : for var in var_list : self . add_slot ( var , " rms " ) @tf . function def _ r e s o u r c e _ a p p l y _ d e n s e ( self , grad , var , apply_state = None ) : coefficients , lr_t , sigma = self . _ get_co effici ents ( grad , var , apply_state ) rms_var = self . get_slot ( var , " rms " ) new_rms = self . alpha * rms_var + (1 -self . alpha ) * tf . square ( grad ) preconditioner = 1./( self . diagonal_bias + tf . math . sqrt ( new_rms ) ) stddev = sigma * tf . math . sqrt ( lr_t * preconditioner ) new_var = var -lr_t * preconditioner * grad + tf . random . normal ( shape = tf . shape ( grad ) , stddev = stddev ) rms_var . assign ( new_rms ) var . assign ( new_var ) Listing 5.2: Implementation of the L-RMSprop optimizer in TensorFlow

  Figure 5.4: Double well potential

Figure 5 . 5 :

 55 Figure 5.5: Performance of preconditioned Langevin algorithms compared with vanilla SGLD on the Adult dataset. The values of the hyperparameters are a = 0.01 and γ = 0.01 for SGLD, L-RMSprop and L-Adam and γ = 0.5 for L-Adadelta, batch size is 32 and the window for the posterior is N = 800. 95% confidence intervals are indicated.

Figure 5 Figure 5 . 7 :Figure 5

 5575 Figure 5.6: Examples of probability predictions for some images from the MNIST dataset

Figure 5

 5 Figure 5.9: Side-by-side comparison of optimization algorithms with their respective Langevin counterparts for the training of a deep neural network on the CIFAR-10 dataset. The values of the hyperparameters are a(n) = A log -1/2 (c 1 n + e) with A = 5.10 -4 and where c 1 n = 1 after 5 epochs; γ n = γ 1 /(1 + c 2 n) where c 2 n = 1 after 5 epochs and where γ 1 = 1 e -3, 2 e -3, 1 e -1 for Adam, RMSprop and Adadelta respectively

  Figure 5.12: Side-by-side comparison of optimization algorithms with their respective Langevin counterparts for the quantization of the law N (0, I 20 ) with 10000 quantization points. The batch size is 512 and each epochs consists in 5 batches. At the end of each epoch, the distortion V is evaluated over 25 × 512 samples and the 95%-confidence intervals are indicated (but may be too small to be visible). The schedules are γ n = 5 e -3 (2 e -2) and a(n) = 5 e -3 (5 e -2) for epochs 0 to 40 and γ n = 5 e -4 (2 e -3) and a(n) = 0 beyond for Adam and RMSprop (for Adadelta).

Figure 5 .

 5 Figure 5.13: Distortion function for µ = N (0, 1) and K = 2.

Figure 5

 5 Figure 5.14: Projection on the first two axes of the quantization of N (0, I 20 ) with 10000 quantization points.

6. 3 .Figure 6 . 1 :Figure 6 . 2 :Figure 6 . 3 :

 3616263 Figure 6.1: Training of neural networks of various depths on the MNIST dataset using Langevin algorithms compared with their non-langevin counterparts. (a): 3 hidden layers, (b): 20 hidden layers, (c): 30 hidden layers, (d): 40 hidden layers. The batch size is 512. The schedules are γ n = 1 e -3 and σ = 5 e -4 for epochs 1 to 12 and γ n = 1e -4 and σ = 0 beyond.

Figure 6 Figure 6 . 5 :

 665 Figure6.4: Layer Langevin method comparison on a dense neural network with 30 hidden layers. The schedules are γ n = 1 e -3 and σ = 5e -4 for epochs 1 to 13 and γ n = 1e -4 and σ = 0 beyond.

Figure 6

 6 Figure 6.6: ResNet elementary residual block

  resnet ' # equals to ' resnet ' or to ' vgg ' ) : self . input_shape = input_shape self . filters = filters self . block_layers = block_layers self . classes = classes self . zero_padding = zero_padding self . mode = mode def identity_block ( self , x , filter_size ) : x_skip = x x = tf . keras . layers . Conv2D ( filter_size , (3 ,3) , padding = ' same ') ( x ) x = tf . keras . layers . B at ch No rma li za ti on ( axis =3) ( x ) x = tf . keras . layers . Activation ( ' relu ') ( x ) x = tf . keras . layers . Conv2D ( filter_size , (3 ,3) , padding = ' same ') ( x ) x = tf . keras . layers . B at ch No rma li za ti on ( axis =3) ( x ) if not self . mode == ' vgg ': x = tf . keras . layers . Add () ([ x , x_skip ]) x = tf . keras . layers . Activation ( ' relu ') ( x ) return x def c o nv o l ut i o na l _ bl o c k ( self , x , filter_size ) : x_skip = x x = tf . keras . layers . Conv2D ( filter_size , (3 ,3) , padding = ' same ' , strides = (2 ,2) ) ( x ) x = tf . keras . layers . B at ch No rma li za ti on ( axis =3) ( x ) x = tf . keras . layers . Activation ( ' relu ') ( x ) x = tf . keras . layers . Conv2D ( filter_size , (3 ,3) , padding = ' same ') ( x ) x = tf . keras . layers . B at ch No rma li za ti on ( axis =3) ( x ) if not self . mode == ' vgg ': x_skip = tf . keras . layers . Conv2D ( filter_size , (1 ,1) , strides = (2 ,2) ) ( x_skip ) x = tf . keras . layers . Add () ([ x , x_skip ]) x = tf . keras . layers . Activation ( ' relu ') ( x ) return x def getModel ( self ) : filter_size = self . filters x_input = tf . keras . layers . Input ( self . input_shape ) x = tf . keras . layers . ZeroPadding2D ( self . zero_padding ) ( x_input )

Figure 6 . 9 :

 69 Figure 6.9: Dense convolutional network block

  Figure 6.10: Layer Langevin method comparison for the training of ResNet-20. The initial number of channels is 16. The schedules are γ n = 1 e -3 (2 e -1 for Adadelta) and σ = 5 e -4(5 e -3 for Adadelta) for epochs 1 to 40 γ n is divided by 10 and σ is set to 0 beyond.

Figure 7 . 3 :Figure 7 . 4 :

 7374 Figure7.3: Example of a trajectory of X t ∈ R 5 along with the controlled fishing u t with N = 100. We recall that the objective biomass is X t ≡ 1.

Figure 7 . 5 :Figure 7 . 6 :

 7576 Figure 7.5: Comparison of Langevin algorithms with their non-Langevin counterparts during the training for the fishing control problem with N = 50 respectively. The schedules are γ n = 2 e -3(5 e -1) and σ n = 5 e -3 (1 e -2) for RMSprop (Adadelta resp.) for epochs 0 to 40 and γ n is divided by 10 and σ n is set to 0 beyond. At the end of each epoch, J is estimated over 50 × 512 trajectories. A zoom on the last epochs for RMSprop is given.

Figure 7 . 7 :Figure 7 . 8 :

 7778 Figure 7.7: Example of trajectory for the deep hedging problem with N = 30.

Figure 7 . 9 :

 79 Figure 7.9: Training of the deep hedging problem with multiple controls with N = 10. The schedules are γ n = 2 e -3(5 e -1) and σ n = 2 e -3 (5 e -3) for Adam and RMSprop (resp. Adadelta) for epochs 0 to 180 and γ n is divided by 10 and σ n is set to 0 beyond.

Figure 7 .Figure 7

 77 Figure 7.10: Example of trajectory for the oil drilling problem with N = 20.

  .3.3) where the constants C and c only depend on the bounds on b Z and σ Z and on their derivatives and their Lipschitz constants, on the modulus of ellipticity of σ Z , on d and on T . Let us also recall the recent result from [MPZ21] giving Aronson's bounds of the partial derivatives with respect to y in the case where b Z is unbounded. Considering [MPZ21, Section 4] with [MPZ21, (3.1)], we have the following result. Theorem 8.3.2. Let Z be the solution of (8.3.1) and let T > 0. Assume Lip t (b Z ) and Lip t

Figure 8 . 1 :

 81 Figure 8.1: Example of trajectory of a Black-Scholes process (8.4.1) with x = 1 and σ = 0.5 along with its one-step Euler-Maruyama scheme (8.4.2).

  B): functions that are k times differentiable with continuous derivatives, • C k b (A, B): functions that are bounded, k times differentiable with continuous and bounded derivatives, • C k b (A, B): functions that are k times differentiable with continuous and bounded derivatives.

0 F

 0 .3.2)9.3.2 Expectation of the supremum of a random path processLemma 9.3.2. Let F (u, s) u≥0,s∈[0,T ] a R-valued random process adapted to the filtration F with respect to its second variable, such that for every s ∈ [0, T ] and u > T ′ , F (u, s) = 0, and such that ∥F ∥ ∞ ≤ C 1 almost surely, ∂ 1 F exists and ∥∂ 1 F ∥ ∞ ≤ C 2 almost surely, for some C 1 , C 2 ∈ R + . and let (M s ) 0≤s≤T be a R-valued martingale adapted to F with E⟨M ⟩ T < ∞. For r ∈ [0, T ] let us define φ u := r (u, s)dM s , u ≥ 0.

  .2.2) We consider the space R 2 endowed with the Euclidean norm denoted by | • |. For D ⊂ R 2 and k ∈ N, we denote by C k b (D) the set of real-valued functions defined on D which are bounded and have bounded partial derivatives up to the order k.

α

  Figure 10.1: Example of a wedge of angle α.

  ∂D -:={(x, y) ∈ D, y = 0} = {(r cos(θ), r sin(θ)), r ∈ R + , θ = 0}, ∂D + :={(x, y) ∈ D, y = ax} = {(r cos(θ), r sin(θ)), r ∈ R + , θ = α}.

Figure 10 . 3 :

 103 Figure 10.3: Partition of R 2 using {D k } k

  Theorem 10.4.2 (Reflected case). Assume that D = ⟨π/m⟩ for some m ∈ N. Then we have the following formulas for the density of the reflected Brownian motion on the wedge D = ⟨π/m⟩: ∀t > 0, x, y ∈ D, P x (X t ∈ dy)

t

  cos(θ 0 -2kα) drdt.

Figure 10 . 4 :

 104 Figure 10.4: Example of the domains of simulation at step n + 1.

Figure 10 . 5 :

 105 Figure 10.5: Example of domains of simulation at the step n + 1 in the reflected case.

  .5.11)

∝

  re -(r-r 0 ) 2 2t drdθ = (r -r 0 )e -(r-r 0 ) 2 2t dr + r 0 e -(r-r 0

  With this inequality and the change of variable u = rr n /(2t ′ ), we obtaind TVn ( XT , X T ) ≤Cε + 8αt ′ Cε + I(ε).The integral I(ε) converges if γ ≡ γ(ε) := 2t ′ /r 2 n > 1, which is satisfied as soon as ε < 2. Then, denoting β := 1 + π/α,

  Brownian motion with constant driftLet b ∈ R 2 and consider the Brownian motion with drift: W t := W t + bt. Then by Girsanov's theorem, W is a Brownian motion under the probabilityQ x 0 |Ft := e -b•Wt-b 2 t/2 P x 0 |Ft , (10.7.1) so that for any bounded test function f : D × [0, T ] → R:

e -b 2 T ∧τ 2 e

 2 b•W T ∧τ f (W T ∧τ ) (10.7.2)

P x 0 0

 0 ( W t ∈ dy, τ > t) = 2r tα e -b 2 t 2 e b•y e -(r 2 +r 2

  Ȳt∧τ := Ȳt k + b( Ȳt k )(t -t k ) + σ( Ȳt k ) • (W t∧τ -W t k ), (10.7.5) This is done using the algorithm in Section 10.7.1. More precisely, we simulate ȲT ∧τ taking b ≡ 0 while keeping track of the Brownian increments; then instead of f ( ȲT ∧τ ), the result of the simulation ( Ȳs ) • b( Ȳs )| 2 ds + T ∧τ 0 (σ -1 ( Ȳs ) • b( Ȳs )) • dW s f ( ȲT ∧τ ). (10.7.6)

Figure 10

 10 Figure 10.6: (left) Histogram of N for the exact reflected algorithm with 50000 Monte Carlo iterations. The horizontal axis represents the values of N while the vertical axis gives the number of trajectories such that N is in the interval noted in the horizontal axis. The last bar counts all the values greater than 150. On the right, average number of iterations in function of ε. The parameters are the same as in Table10.1 and for each value of ε, 50000 Monte Carlo iterations are used.

P

  ( X ∈ dx| Y = y) = P(X ∈ dx| Y = y) if y ∈ A P(X ∈ dx| Y / ∈ A) if y / ∈ A Then: P( X ∈ dx) = P( X ∈ dx| Y = y)P(Y ∈ dy) = y∈A P(X ∈ dx| Y = y)P(Y ∈ dy) + y / ∈A P(X ∈ dx| Y / ∈ A)P(Y ∈ dy) = P(X ∈ dx, Y ∈ A) + P(X ∈ dx| Y / ∈ A)P(Y / ∈ A) = P(X ∈ dx).

π π 0 e:2

 0 x) ≤ e x + 1 π(ν + x) .(10.9.2)Proof. The inequality follows from the integral representation [Wat44, p.181]:I ν (x) = 1 x cos θ cos(νθ)dθsin(νπ) π ∞ 0 e -x cosh(t)-νt dt.Now we provide two results on estimates related to the law of F (T ) = T Let (R t ) t≥0 be a Bessel process of dimension 2, or equivalently, of index 0. ThenP r 0 [F (T ) ≥ x]We use[START_REF] Jeanblanc | Mathematical methods for financial markets[END_REF], Proposition 6.2.5.1, with a = 0 and ν = 0:E r 0 e -Performing the change of variable w := T v 1+2vT , so that v = w T (1-2w) , we have for all λ > 0:E r 0 e -λ 2 2 F (T ) = r λ 0 Γ(λ/2)T λ/2 -1 (1 -2w) λ/2 e -r 2 0 w/T dw.

  1 + I 2 .

  But we have, for all a ≥ 0: if a ≥ 1, e -1 + log(1/a) if a < 1.1 log(1 -u 2 )du,where the last integral converges as soon as p < 2.10.10 Appendix: Proofs in Section 10.4Proof of Theorem 10.4.2 in the case that D = ⟨π/m⟩. We apply the method of images in R 2 . Let f : D → [0, +∞) be a non-negative continuous function with compact support. The heat equation with boundary conditions∂ t u(t, x) = 1 2 ∆u(t, x), (t, x) ∈ [0, +∞) × D u(0, x) = f (x), x ∈ D ∇u(t, x) • n(x) = 0, t > 0, x ∈ ∂D, (10.10.1)where n(x) denotes the inward unitary orthogonal vector on the boundary, can be rewritten as a partial differential equation in polar coordinates (r, θ) as described in [MNP00, Chapter 1]. Furthermore if we perform the change of variable z = log(r), we obtain a parabolic problem with non-constant coefficients in a strip with mixed type boundary condition. The existence and uniqueness for this PDE is treated in [LSU68, chapter IV, Theorem 5.3]. From this reference and the Feyman-Kac representation theorem we obtain thatu(t, x) := E x [f (X t )]satisfies the heat equation on D described above. Now we define the functionf : R 2 → R y → f (T -1 k y) for y ∈ D k ,and for t ≥ 0 and x ∈ R 2 , ũ(t, x) := E x f (W t ) .Then ũ satisfies the heat equation on R 2 with f as initial condition, so that ũ(t, x) = 1 2πt R 2 f (y)e -|x-y| 2

  (x -T k y) • n(x) = -(x -T 2m-1-k y) • n(x), |x -T k y| 2 = |x -T 2m-1-k y| 2 , so that ∇ũ(t, x) • n(x) = -1 2πt 2 D f (y) 2m-1 k=0 (x -T k y) • n(x)e -|x-T k y| 2 T k y) • n(x)e -|x-T k y| 2 T 2m-1-k y) • n(x)e - |x-T 2m-1-k y| 2 2t dy = 0.If x ∈ ∂D + , we get the same result noting that for all y ∈ D and k = 0, . . . , 2m -1,(x -T (k+1) mod 2m y) • n(x) = -(x -T (2m-k) mod 2m y) • n(x), |x -T (k+1) mod 2m y| 2 = |x -T (2m-k) mod 2m y| 2 ,So u(t, x) = ũ(t, x) for all t ≥ 0 and x ∈ D, andE x [f (X t )] = 1 2πt D f (y)2m-1 k=0 e -|x-T k y| 2 2t dy.

t

  cos(θ 0 -ϑ k ) drdθ. (10.10.3)We use the following identity (see [GR07, page 933, (8.511.4)]), valid for γ, z ≥ 0:e γz = I 0 (z) + 2 ∞ n=1 T n (γ)I n (z),where T n is the n th Tchebychev's polynomial of the first kind and I n the modified Bessel function of the first kind with order n. Then the result (10.3.4) follows because T n (cos(θ)) = cos(nθ) and2m-1 k=0 cos(n(θ 0 -ϑ k )) = 2m cos(nθ) cos(nθ 0 ) if n is a multiple of m, 0 otherwise.Using the above properties and m = π/α, we obtain P x (X t ∈ dy)

  θ)m n (θ 0 ) drdθ.(10.10.5)

  Proposition 10.11.1. Let (R t ) t≥0 be a Bessel process in dimension d ≥ 3 and let F (T ) = t 0 ds/R 2 s . Then: E r 0 [F (T )] < +∞.

0 I

 0 ν (ur 0 )e -εu 2 /2 u ν-1 du < ∞,

π π 0 e 0 e

 00 0 , θ 0 ), (10.12.2)where f : [0, ∞) × [0, α] → R denotes the function f expressed in polar coordinates.For the proof, we use the following representation:∀x > 0, ∀α ≥ 0, I α (x) = 1 x cos(u) cos(αu)dusin(απ) π ∞ -x cosh(u)-αu du.

e

  -rr 0 t cosh(u)-nπ α u dudrdθ.

  )e -nb = ℑ ∞ n=0 e ina-nb = e -b sin(a) (1 -e -b cos(a)) 2 + e -2b sin 2 (a) . (10.12.3) We use the following trigonometric identity written in compact form 4 cos(a) cos(b) sin(c) = sin(a + b + c) + sin(c -a -b) + sin(a -b + c) + sin(c -a + b)(10.12.4)A jk (θ) := π α ((-1) j θ + (-1) k θ 0 + π),

e

  -rr 0 t cosh(u) g A jk (θ),

,

  g(a, b) = e -b sin(a) (1 -e -b cos(a))2 + e -2b sin 2 (a)∼ b 2 ) -log(b 2 ) db < ∞. Next, note that ∂ b g(a, b) is negative, so ∀u ≥ 1, ∀θ ∈ R, g A jk (θ),and since u → u 2 / cosh(u) is non-negative and bounded above, there exists B > 0 such that

g

  N (a, b) := ∞ n=N sin(na)e -nb = ℑ ∞ n=N e -nb+ina = e -N b sin(N a) -e -b sin((N -1)a)

e

  -rr 0 t cosh(u) g N A jk (θ), that e N b g N (a, b) = sin(N a) -e -b sin((N -1)a) (1 -e -b cos(a)) 2 + e -2b sin 2 (a) ∼ a,b→0 a a 2 + b 2 , so that (a, b) → e N b g N (a, b

  Lemma 10.12.3 for the above convergence. Moreover for all u ≥ 1, θ ∈ R, |g N (A jk (θ),

Lemma 10.12. 3 . 0 f

 30 Let b > 0 and let f ∈ L 1 ((0, b)) be non-negative and continuous. Then: Let ε > 0 and choose δ > 0 such that δ 0 f (x)dx ≤ ε. Then for N big enough: b 0 e -N x f (x)dx ≤ δ (x)dx + e -N δ ||f || 1 ≤ 2ε.

  nπ(θ 0 + u) α + cos nπ(θ 0 -u) α α 0 f (r, θ) cos nπθ α dθ.

  θ 0 ± u).

Case 1 :

 1 If r 0 cos(u) > 0:re -(r-r 0 cos(u)) 2 2t √ t = 1 √ t (r -r 0 cos(u))e -(r-r 0 cos(u)) 2 2t + r 0 cos(u) e -(r-r 0 cos(u)) 2 2t √ t .The total mass of the first term is r 0 cos(u))e -(r-r 0 cos(u)

  0 cos(u), θ 0 ± u)e - δ > 0 and take ε small enough such that cos(u) ≃ 1 and sin(u) ≃ u for all u ∈ [0, ε].

  r, θ 0 + u) + f (r, θ 0 -u))dr.

f 2 (u) := α 4 √√ 0 f 1 0 f 1

 40101 2πr 0 cos(u) f (r 0 cos(u), θ 0 + u) + f (r 0 cos(u), θ 0 -u) , and in Step 3 that π 0 f 1 (u, t)f 2 (u)du converges when t → 0 to lim t→0 2πr 0 ( f (r 0 cos(u), θ 0 + u) + f (r 0 cos(u), θ 0 -u))e - r 0 , θ 0 ) = f (r 0 , θ 0 ).To end the proof of the convergence, we have to show that lim t→0 π (u, t)f 2 (u, t)du = lim t→0 π (u, t)f 2 (u)du.

trn 2 π 2

 2 cos(u) | fN (r, θ 0 + u) -f (r, θ 0 + u)|du dr where fN denotes the N th partial Fourier sum of f. Then, by Cauchy-Schwarz inequality: ,θ 0 + u) -f (r, θ 0 + u) || fN (r, •) -f (r, •)|| 2 ||∂ θ f || ∞ .

  Figure 1.8: Architecture of the VGG-16 network for an input image of size 224 × 224.

	Input layer	3 × 3 conv, 64 channels	3 × 3 conv, 64 channels	2 × 2 pool	3 × 3 conv, 128 channels	3 × 3 conv, 128 channels	2 × 2 pool	3 × 3 conv, 256 channels	3 × 3 conv, 256 channels	3 × 3 conv, 256 channels	2 × 2 pool	3 × 3 conv, 512 channels	3 × 3 conv, 512 channels	3 × 3 conv, 512 channels	2 × 2 pool	3 × 3 conv, 512 channels	3 × 3 conv, 512 channels	3 × 3 conv, 512 channels	2 × 2 pool	flatten	dense, 4096 units	dense, 4096 units	dense, classes units
	Size:224 × 224		Size:112 × 112 Size:56 × 56		Size:28 × 28		Size:14 × 14		Size:1

  Figure 1.9: Depth of Markovian neural networks. The control u acts on Y t k , which itself acts on Y t k+1 , Y t k+2 , . . ., Y t N , hence the depth of the network. International Neural Network Society Workshop on Deep Learning Innovations and Applications (INNS DLIA), part of the Machine Learning conference International Joint Conference on Neural Networks IJCNN 2023, and will be published in the first edition of the INNS workshop series in Procedia Computer Science as Langevin algorithms for very deep Neural Networks with application to image classification.

usable as the Python package langevin_optimizers, which is downloadable from the GitHub repository https://github.com/Bras-P/langevin-for-stochastic-control and with the command 1 $ pip install git + https :// github . com / Bras -P / langevin -for -stochasticcontrol . git

These results have been presented at the

  Langevin-based algorithms are used to solve optimization problems in high dimension and have gained much interest in relation with Machine Learning [WT11, MCF15, LCCC16, DM17]. The Langevin equation is a Stochastic Differential Equation (SDE) which consists in a gradient descent with noise. More precisely, let V : R d → R + be a coercive potential function, then the associated Langevin equation reads

Theorem 2.2.1. (

	a) Let Y be defined in (2.2.5). Assume (2.2.2, H V 1 ), (2.2.8, H V 2 ), (2.2.9, H σ ),
	(2.2.10) and (2.2.11, H cf ). If A is large enough, then for every x 0

Table 2

 2

	Adadelta

.1: Best accuracy performance on the MNIST test set after 10 epochs.

  ) .

	• Let us now investigate d TV ( Xx,n t , Ȳ x t,u ). Conditionally to ζ(x), both random vectors are
	Gaussian vectors with
	Xx,n t

  ||h|| 2 , 4.5. Proofs of Theorem 4.3.3 and Theorem 4.3.5 using Theorem 4.4.1 because for all i, α i ≤ 1 2 . If t and h are such that (t α 1 h 1 , . . . , t α d h d ) ∈ B(0, δ), then choosing δ such that for all (t α 1 h 1

  use the Machine Learning library TensorFlow [AAB+ 15] with Python. Neural networks can be easily built by stacking layers:

	import tensorflow as tf
	model = tf . keras . Sequential ([
	tf . keras . layers . Conv2D (8 , (4 , 4) , activation = ' sigmoid ') , # 8 filters
	with convolution kernel of size 4 x4
	tf . keras . layers . MaxPooling2D ((2 , 2) ) , # Pooling with infinity norm on
	2 x2 squares
	tf . keras . layers . Flatten () ,
	tf . keras . layers . Dense (32 , activation = ' sigmoid ') , # 32 units
	tf . keras . layers . Dense (10)
	])
	Listing 5.1: Example of a 2D-convolutional neural network in TensorFlow
	Once the model is built, we compile it with a loss function and an optimizer:

  Performance of preconditioned Langevin algorithms compared with vanilla SGLD on the JENA weather dataset. The values of the hyperparameters are a

												SGLD
	Test accuracy	1 1.5								L-RMSprop L-Adam L-Adadelta
		0.5
		0	1	2	3	4	5	6	7	8	9	10	11
							Epochs
	Figure 5.11:
								.2, showing that preconditioned optimizers achieve
	a faster convergence speed.
		Preconditioner SGLD L-RMSprop L-Adam L-Adadelta Baseline model
		Best test loss	0.739		0.434		0.433	0.467		0.545
		Table 5.2: Best accuracy performance on the JENA weather test set after 10 epochs.

  Bayesian setting and compare their performances with non-Langevin stochastic gradient algorithms. As it was noted in [NVL + 15, Ani19], adding gradient noise can in fact improve the learning. Similarly, noisy activation functions [GMDB16, SLH + 19] may yield better learning for very deep neural networks. Indeed, the noise provides regularization and allows to escape from traps for the gradient descent such as local minima and saddle points [DPG + 14]. Moreover, the deeper the neural network is, the more non-linear it is, thus increasing the number of such traps. Nonconvex optimization through Langevin algorithms shares heuristics with simulated annealing which consists in sampling with respect to a Gibbs measure where the noise parameter gradually decreases to zero, see[START_REF] Van Laarhoven | Simulated annealing: theory and applications[END_REF] and Chapter 2.Many advances in supervised learning were made possible using very deep neural networks, which are able to tackle much more difficult problems than shallow ones [KSH12,[START_REF] Montúfar | On the number of linear regions of deep neural networks[END_REF][START_REF] Lecun | Deep learning[END_REF], in particular as it comes to image classification [SLJ + 15, SZ15, HZRS16, HLVDMW17]. Still, deep neural networks which consist in a succession of dense layers are considerably more difficult to train [GB10, DPG + 14] and may run into vanishing gradient problems

Langevin algorithms are widely used for the training of neural networks in a Bayesian setting [WT11, VBB

+ 20]

. Adding a small exogenous noise adds regularization to the training and allows to quantify the degree of uncertainty on the parameters. In this paper, we consider Langevin algorithms directly used for stochastic optimization of neural networks in a non-

  now prove Theorem 8.2.2. For the proof of Theorem 8.2.5, we do not use Lemma 8.3.7; instead we directly apply Theorem 8.2.7. Using Theorem 8.3.2, ∇ k y p X and ∇ k y p Y exist for k = 0, 1, . . . , 2r and satisfy the same bounds as previously. Then using Theorem 8.2.7 with Lemma 8.5.3 we obtain Proof. We use Lemma 8.3.7 again and rework the bound on d TV (Z x t , V x t ) by paying attention to the dependency of the constants in r in the proof of Theorem 8.2.7 with Z 1 := Z x t and Z 2 := V x t . Since σ 1 ∈ C 2r b for every r ∈ N, we write (8.3.24) for any r ∈ N and we have

	8.3.5 Proof of Theorem 8.2.5
		d TV (X x t , Y x t ) ≤ C(	√	t(1 + ∆b(x)) + ∆σ(x) + t|b(x)|) 2r/(2r+1) .
	8.3.6 Proof of Theorem 8.2.8
	Proof. Using Lemma 8.3.7, we have
		d TV (X x t , Y x t ) ≤ Ce C|x| 2 t 1/2 + d TV (Z x t , V x t )	(8.3.32)
	We now apply Theorem 8.2.7 with the random vectors Z 1 = Z x t and Z 2 = V x t . Assuming that σ 1 is C 2r b and using Theorem 8.3.1, ∇ k y p Z exists for k = 0, 1, . . . , 2r and
	∀k = 0, 1, . . . , 2r, ∀t ∈ (0, T ], ∀x, y ∈ R d , ∥∇ k y p Z (0, t, x, y)∥ ≤	C t (d+k)/2 e -c|y-x| 2 /t .
	Then we have
	R d	∇ 2r y p Z (0, t, x, ξ)dξ ≤ Ct -r	R d	1 t d/2 e -c|x-ξ+y| 2 /t dξ ≤ Ct -r .
	The same way we have
		R d	∇ 2r y p V (0, t, x, ξ)dξ ≤ Ct -r .
	Applying Theorem 8.2.7 with Lemma 8.5.3 yields
		d TV (Z x t , V x t ) ≤ C(	√	t + ∆σ(x)) 2r/(2r+1) .

  2 . Let X and Y be the solution to the two general SDEs (8.2.1) and (8.2.2). Assume that b and σ are Lipschitz continuous in (t, x) and that σ is bounded. Then for every p ≥ 1,

	8.5. Appendix
	Lemma 8.5.3. ∀t ∈ [0, T ], ∀x ∈ R d , ∥X x t -Y x t ∥ p ≤ C t(1 + ∆b(x)) + t 3/2 (|b 1 | + |b 2 |)(0, x) + ∆σ(x)t 1/2
	(8.5.1)

  .2.4) This scheme more convenient to simulate as it only requires the simulation of the Brownian increments(W t k+1 -W t k ) 0≤k≤N -1 .With no ambiguity, in the proofs we shall use the notation ψ for ψ some function defined on R d , such that for every process Y and every s ∈ [0, T ] we have ψ(Y s ) = ψ(Y

	s).

  T ], Y t (u) is well defined and Y t : u → Y t (u) is continuous. Moreover, for every t ∈ [0, T ], Y t is F t -measurable (but the process (Y t (u)) u≥0 is not adapted w.r.t. u) and using Lemma 9.3.2 again, Y t is almost surely C 1 and Lipschitz-continuous with Ẏt (u) =

			.4.1)
	Following Lemma 9.3.2, for every u ≥ 0 and t ∈ [0, t	t
	∂ 1 K 1 (t + u, s)b(X s )ds +	∂ 1 K 2 (t + u, s)σ(X s )dW s ,	(9.4.2)
	0	0

  we have indeed P r,t : Ω 2T → Ω 2T . Then following (9.4.4) and (9.4.5) we have

					P r,t (Y t ) = Y t+r .			(9.4.8)
	Likewise, we define
	Ȳt (u) =	t	K 1 (t + u, s)b(	Xs	)ds +	t	K 2 (t + u, s)σ(	Xs	)dW s , u ≥ 0, t ∈ [0, T ]	(9.4.9)
	0				0
	as well as the semi-group corresponding to the Euler-Maruyama scheme (9.2.3) for k ∈ {0, . . . , N -
	1} and r ∈ [0, T -t k ]:

  Figure 10.2:Table summing up the four different cases of wedge D

	Cartesian	{y ≥ 0 and y ≤ ax},	{y ≥ 0 and y ≥ ax},	{y ≥ 0 or y ≥ ax},	{y ≥ 0 or y ≤ ax},
	Equation	a > 0	a < 0	a > 0	a < 0
	Value of α	arctan(a)	π + arctan(a)	π + arctan(a)	2π + arctan(a)
	Wedge for
	the original problem	α	α	α	α
	a a a a
	Equation of W ′ if σ2 -aσ1ρ > 0	{y ≥ 0 and y ≤ a ′ x}, a ′ > 0	{y ≥ 0 and y ≥ a ′ x}, a ′ < 0	{y ≥ 0 or y ≥ a ′ x}, a ′ > 0	{y ≥ 0 or y ≤ a ′ x}, a ′ < 0
	Equation of W ′ if σ2 -aσ1ρ < 0	{y ≥ 0 and y ≥ a ′ x}, a ′ < 0	{y ≥ 0 and y ≤ a ′ x}, a ′ > 0	{y ≥ 0 or y ≤ a ′ x}, a ′ < 0	{y ≥ 0 or y ≥ a ′ x}, a ′ > 0
		W

  := Zt k+1 + (W t -W t k ) + L 2 t -L 2 t k .(10.7.11)

	L 1 t k	(10.7.10)
	Zt+t

k

  Zt i+1 ) -u(t i , Zt i ) +The following step relies on using the Taylor expansion for each of the terms involving ∇u(t i , Zs ) and (∂ t + (1/2)∆) u(s, Zs ) at (t i , Zt i ) together with the fact that u solves (10.7.14). Without going into much detail, let us consider the expansion of the term ∂

	t i+1 t i	∂ t +	1 2	∆ u(s, Zs )ds
	t i+1
	=
	t

i E x 0 b( Zt i )∇u(t i , Zs ) + ∂ t + 1 2 ∆ u(s, Zs ) ds. t u(s, Zs ): ∂ t u(s, Zs ) = ∂ t u(t i , Zt i ) + 1 0 ∇∂ t u(t i , α Zs + (1 -α) Zt i ) • ( Zs -Zt i )dα + 1 0 ∂ t ∂ t u(αs + (1 -α)t i , Zs ) • (s -t i )dα

  10.8. SimulationsE x 0 95 % interval Time (s) MC iterations E x 0 [N ] E x 0 [f (W τ )] with [Met09] 3.473Table10.1: Simulations with α = 0.9, r 0 = 1.5, θ 0 = 0.3, f (x, y) = x 2 + y 2 , T = 1.

			± 0.082	0.67	50000	1
	E x 0 [f (W τ )]	3.489	± 0.085	1.04	50000	1.45
	E x 0 [τ ] with [Met09]	0.742	± 0.030	7.06	20000	-
	E x 0 [τ ]	0.590	± 0.024	4.92	20000	-
	E x 0 [f (W τ ∧T )]	2.980	± 0.049	5.79	10000	1.37
	E x 0 [(W τ ∧T ) • e 1 ]	1.441	± 0.012	6.26	10000	1.37
	E x 0 [f (X T )], ε = 0	4.138	± 0.224	13.2	1000	75.6
	E x 0 [f (X T )], ε = 0.03	4.313	± 0.072	28.2	10000	5.11
		E x 0	95 % interval Time (s) MC iterations E x 0 [N ]
	E x 0 [f (W τ ∧T )]	0.195	± 0.003	8.85	10000	1.28
	E x 0 [f (X T )], ε = 0.03 0.117	± 0.003	9.26	5000	2.73
	Table

Table 10

 10 Table 10.1 and for each value of ε, 50000 Monte Carlo iterations are used. E x 0 95 % interval Time (s) MC iterations Discretization stepE x 0 [f (Y τ ∧T )] [Girsanov] 2.83

			± 0.06	456	5000	T/500
	E x 0 [f (Y τ ∧T )] [Two-step] 2.84	± 0.06	457	5000	T/500
	E x 0 [f (Z T )], ε = 0.01	3.77	± 0.09	1140	5000	T/500

.3: Simulation of the process dY t = -µ(Y t -κ) + σdW t

  1-p . (10.9.6) Proof. By [JYC09, (6.2.3)], the density of R t in r is

	r t	r r 0	ν	e -	r 2 +r 2 0 2t	I ν	rr 0 t	,	(10.9.7)

Our code is available at https://github.com/Bras-P/langevin-simulated-annealing.

An implementation of Langevin optimizers in TensorFlow is available at https://github.com/Bras-P/ deep-layer-langevin.

We remark that max(q, K) = K -ReLU(-q + K).

The cases where a = 0 are not considered here because they lead to cases where the problem can be simplified to a one dimensional problem.

We choose the angle of the process (θ(Bt)) t≥0 so that it is continuous with respect to t.

The programs with a demonstration notebook are available at https://github.com/Bras-P/simulation_ reflected_brownian_motion_wedge

Remerciements

Je remercie tout d'abord mon directeur de thèse, Gilles Pagès, pour m'avoir accepté en tant que doctorant et pour m'avoir guidé et accompagné pendant ces trois années. J'ai beaucoup appris sous sa direction et ses précieux conseils et remarques m'ont beaucoup aidé à prendre du recul et à organiser mes recherches. Tout au long de ma thèse, j'ai été impressionné par ses larges connaissances et son expertise dans de très nombreux aspects des probabilités numériques et appliquées, et il avait (presque) toujours réponse à mes questions. Je me souviens de tous les moments où je toquais à sa porte à l'improviste et malgré toutes ses responsabilités et occupations, mon professeur trouvait toujours un moment de libre pour discuter.

Je remercie ensuite les rapporteurs de cette thèse Éric Moulines et Mike Giles, pour avoir accepté cette charge. Leur rapport et leurs commentaires pertinents m'ont grandement aidé à améliorer la qualité de ce manuscrit. Je remercie également les autres membres du jury, Francis Bach, Olivier Pironneau, Gersende Fort, Nicolas Fournier et Josef Teichmann, pour l'honneur qu'ils me font en participant au jury de ma soutenance. Leur expertise et leur expérience dans leur domaine sont extrêmement précieuses pour évaluer mes travaux.

Remerciements conscience que l'apprentissage machine et la simulation numérique sont des domaines très larges impliquant et mélangeant un large éventail de compétences et de spécialistes.

Acknowledgements

I would like to thank Gilles Pagès for insightful discussions.

Acknowledgements

I would like to thank Gilles Pagès for insightful discussions.

Acknowledgements

The authors thank Idris Kharroubi for helpful discussions.

Acknowledgements

The authors thank Professor Gilles Pagès, Sorbonne University, Paris, and Professor Toshihiro Yamada, Hitotsubashi University, Tokyo, for insightful discussions. The first author thanks the Japanese Society for Promotion of Science JSPS for financial support for a visit to Osaka University and to Hitotsubashi University within the JSPS Summer Program.

Acknowledgements

The first author thanks École Normale Supérieure, Département de Mathématiques et Applications for providing a financial support for a visit to Ritsumeikan University.

The second author is supported in part by KAKENHI 20K03666. Both authors thank the anonymous referee for a careful review and for correcting mistakes and providing references.

Appendix: Properties of tensors

Proposition 4.8.1. Let T k be a symmetric tensor of order k in R d . Let E be a subspace of R d . Assume that ∀h ∈ E, T k • h ⊗k = 0.

Then we have ∀h 1 , . . . ,

.1: Depth of Markovian neural networks. The control u acts on X t k , which itself acts on X t k+1 , X t k+2 , . . ., X t N , hence the depth of the network.

where J is a subset of weight indices. In particular, we denote LL-name p% the Layer Langevin name algorithm where the Langevin layers are chosen to be the first p% layers.

Experimental setting

We proceed to side-by-side comparison of Langevin algorithms (7.2.9) with their non-Langevin counterparts (7.2.8) on various SOC problems. We consider a first case where the control is given by only one neural network depending on t and a second case where a different neural network is used for each control time (7.2.4). In this second case, since we can expect two consecutive control networks to have close parameters, one usual way of performing the training procedure is to first train networks for a small amount of control times, then to perform the whole training through transfer learning. We do not expect Chapter 8

Total variation distance between two diffusions in small time with unbounded drift: application to the Euler-Maruyama scheme

Chapter 9

Weak error rates for numerical schemes of non-singular Stochastic Volterra equations with application to option pricing under path-dependent volatility

The results presented in this chapter are a joint work with Masaaki Fukasawa. They have been submitted to SIAM Journal on Financial Mathematics (SIFIN ) and are currently in revision for possible publication.

Abstract

We study the weak error rate for the Euler-Maruyama scheme for Stochastic Volterra equations (SVE) with application to pricing under stochastic volatility models. SVEs are non-Markovian stochastic differential equations with memory kernel. We assume in particular that the kernel is non-singular and C 4 . We show that the weak error rate is of order O(1/N ) where N is the number of steps of the Euler-Maruyama scheme, thus giving the same weak error rate as for SDEs. Our proof consists in adapting the classic weak error proof for Markov processes to SVEs; to this end we rely on infinite dimensional functionals and on their derivatives. Our work opens the way to study rough SVEs, which shall be investigated in a next paper.

Remark 9.3.3. In the following, we will use Lemma 9.3.2 for families of trajectories (φ t ) t∈[0,T ] of the form

When the assumptions of Lemma 9.3.2 are checked, the bound (9.3.3) is true up to some modification of φ, i.e. for a family of trajectories (

Without loss of generality, each time we use Lemma 9.3.2 we do not mention explicitly the modification.

A general Itō formula for path-dependent functionals

In this section we prove an extension of the classic Itō formula to processes of the form G(φ t ), where G : Ω T ′ → R and where for every t ∈ [0, T ], φ t is some F t -measurable random path.

Theorem 9.3.4. Let us consider the family of random paths (φ t u ) t∈[0,T ], u≥0 such that

where for every u ≥ 0, s

and such that Z i (u, s) = 0 for u > T ′ , T ′ ∈ R + . We also assume that φ 0 ∈ Ω T ′ ∩ C 1 and is Lipschitz-continuous. Moreover, let G : Ω T ′ → R with bounded pathwise derivatives up to order 3. Then we have almost surely

Remark 9.3.5. We highlight the fact that in (9.3.4), the values of Z i (u, s) cannot depend on t.

For example, if we consider the SVE

for some adapted semi-martingale A, then we need to write φ t u as

Proof. We first remark that if G only depends on a finite number of times u 1 , . . . ,

The estimate for the weak error in small time (9.4.14) also holds for |( ⇀ P h,t k -P h,t k )g(ω)|; the only necessary adaptation in the proof is for the estimate for I 2 and I 3 . Indeed, instead of (9.4. 19) we need to prove

and since g, K 2 , ∂ 2 K 2 and σ are bounded we have

The argument for the estimate of I 3 is similar.

Having proved the estimate for the weak error in small time, the conclusion of the proof is the same as for X.

Simulations

In order to numerically check the convergence rate obtained in Theorem 9.2.2, we empirically measure the weak convergence rate in the case of a stochastic volatility model where the volatility follows some Volterra equation. We consider the following Volterra version of the Stein-Stein model [START_REF] Stein | Stock price distributions with stochastic volatility: An analytic approach[END_REF] where the analogous rough version was introduced in [AJ22]:

where the asset price process S and the square volatility process V take their values in R, the function g 0 : R + → R is deterministic and continuous, the processes B and W are standard Brownian motions with correlation ρ ∈ [-1, 1] and the non-singular kernel K is given by

The process (S t , V t ) ⊤ in (9.5.1) is a special case of the Volterra equation (9.2.1) with 2 × 2 matrix kernels We consider the payoff function given by the Call option

We simulate ( ⇀ S T , ⇀ V T ) by discretizing the kernel K using weights matching the second mo- ment, see [Gas23, Section 3] and we plot Ef ( ⇀ S ⌊βN ⌋ T ) -Ef ( ⇀ S N T ) for some β ∈ (1, 2] and for different values of N , where N is the number of steps in the Euler-Maruyama scheme of the

An example of trajectory is given in Figure 9.1 and the results are given in Figure 9.2 with the following parameters: T = 1, S 0 = 1, K = 1, κ = 0.01, X 0 = 0.1, ρ = -0.7, ν = 0.05, A 1 = 0.3, A 2 = 0.02, β = 1.5, g 0 : t → (4θ)/(3A 1 )t 3/4 , θ = 0.01.

We empirically obtain a convergence rate for the weak error which is approximatively -1, thus confirming the results in Theorem 9.2.2.

Appendix

We use the following version of the Kolmogorov continuity theorem, giving the precise upper bound constant.

Theorem 9.6.1 (Kolmogorov continuity theorem). Let (X t ) t∈[0,T ] be a R d -valued random process and assume that for some p, ϵ > 0,

Chapter 10

Simulation of Reflected Brownian motion on two dimensional wedges

A slight generalization of (10.3.3) to the wedge V = ⟨α -, α + ⟩ with

gives, for y ∈ V:

dy, (10.5.9)

where T k (r cos(θ), r sin(θ)) = (r cos( ϑ k ), r sin( ϑ k )). We then apply acceptance-rejection sampling with the reference density proportional to

and accept the sample y under the condition

Algorithm for the simulation of the stopped Brownian motion: General case

In this subsection, we give a recursive algorithm to simulate the final value W T ∧τ of the stopped process on the wedge D, for any angle α ∈ (0, 2π). This algorithm will be used for the simulation algorithm of the reflected Brownian motion, but it also has its own interest as it can be used as itself for the simulation of the stopped Brownian motion. In what follows, we denote F n the sigma algebra generated by the algorithm until the step n, i.e. F n = σ(τ 1 , y 1 , . . . , τ n , y n ), where τ 1 , . . . , τ n and y 1 , . . . , y n are random variables generated by the algorithm. We also denote by P n and E n the conditional probability and expectation with respect to F n . First, we propose an algorithm for the case where the angle is π/m. At the step n of the algorithm we will define a wedge V n ⊂ D of angle π/m for some m ∈ N and then simulate the process stopped on this wedge. Define Θ := max ({π/m, m ∈ N} ∩ (0, α]). Note that since α ∈ (0, 2π), we have α/2 < Θ ≤ α.

Algorithm I: Start at the point x 0 = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) =: y 0 at time T 0 = 0. The algorithm follows these steps for n ∈ N:

1. Given y n = (r n cos(θ n ), r n sin(θ n )) ∈ V n , define the angles:

By the definition of Θ, we have Θ ≤ α and therefore 0 ≤ Θ/2 ≤ α -Θ/2 ≤ α.

Consider the wedge

Proof. Fix K ∈ N. For a wedge V n , n ≤ N in Algorithm I, we say that "V n intersects the boundary of D" if β + n = α or β - n = 0. For n ≤ N , we have two possible cases: • V n does not intersect the boundary of D. Then, using the fact that Θ > α 2 and the definition of β ± n+1 , V n+1 intersects the boundary of D, and θ n is closer to β ± n+1 than β ∓ n+1 , where here the sign ± is such that ∂V ± n+1 ∩ ∂D ̸ = {0}. So using (10.5.5), P n (y n+1 / ∈ ∂D) ≤ 1/2.

• V n intersects the boundary of D. Then if θ n+1 / ∈ {0, α}, V n+1 does not intersect the boundary of D and by the same reasoning as above P n (y n+2 / ∈ ∂D) ≤ 1/2.

Then, by tower property of conditional expectations, we have:

So that:

Algorithm for the simulation of the reflected Brownian motion

We now state an algorithm to simulate the final value of the reflected Brownian motion X T in a wedge D = ⟨α⟩ of any angle α ∈ (0, 2π).

Algorithm II: Choose Θ := max ({π/m, m ∈ N} ∩ (0, α]) as before. Start at the point x 0 = (r 0 cos(θ 0 ), r 0 sin(θ 0 )) =: y 0 at time T 0 = 0. In general, suppose that for n ∈ N the point y n = (r n cos(θ n ), r n sin(θ n )) has already been simulated.

1. Define the angles:

2. Consider the wedge V n+1 := ⟨β - n+1 , β + n+1 ⟩. Note that although y n ∈ V n+1 , we do not necessarily have that V n+1 ⊂ D.

3. Simulate the random variables τ n+1 and z n+1 , which are respectively the time and the final point of a Brownian motion Z = (Z t ) t≥0 starting at y n at time 0 and stopped on the wedge V n+1 using the steps 3-5 of Algorithm I in Section 10.5.2. Note that, since θ n is in the middle of the wedge V n+1 , we have by symmetry P zn (Z τ n+1 ∈ ∂V ± n+1 ) = 1/2.

4. If T n + τ n+1 < T , then we define r n+1 and θ n+1 to be respectively the radius and the angle of z n+1 . Since it is possible that z n+1 / ∈ D, we define θ n+1 as follows:

if θ n+1 = β - n+1 and β - n+1 < 0, 2α -θ n+1 if θ n+1 = β + n+1 and β + n+1 > α. And then we define y n+1 := (r n+1 cos(θ n+1 ), r n+1 sin(θ n+1 )), so that y n+1 ∈ D. Note that if z n+1 / ∈ D, then y n+1 is the reflection of z n+1 with respect to the line {θ = 0} in the second case or {θ = α} in the third case in the definition of θ n+1 . Then define the time:

and the algorithm iterates.