Manon Costa
email: manon.costa@math.univ-toulouse.fr

Sébastien Gadat
email: sebastien.gadat@tse-fr.eu

Non Asymptotic

NON ASYMPTOTIC CONTROLS ON A RECURSIVE SUPERQUANTILE APPROXIMATION

Keywords: September 28, 2020. 2010 Mathematics Subject Classification. Primary: 62L20, Secondary: 60F05, 62P05 Stochastic approximation, Quantile and Superquantile, Non-asymptotic controls, Diffusion approximation

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1. Introduction 1.1. Quantiles and superquantiles. In this paper, we are interested in the estimation of a standard financial risk measure with a recursive stochastic algorithm. Let's be given a real random variable X that mimics the outcome of a portfolio of some financial assets, [ADEH99] introduces a set of axioms that shall describe in actuarial science and financial economics a coherent risk measure (denoted by ρ in this introduction). One of these key properties is that ρ must bring the diversification principle: the risk of two portfolios together is less than the two individual risks, which means from a mathematical point of view that for two random variables Z 1 and Z 2 , ρ satisfies ρ(Z 1 + Z 2 ) ≤ ρ(Z 1 ) + ρ(Z 2 ). In some recent works, this subadditive property has been relaxed and replaced by a convex axiom: ρ(λZ 1 + (1 -λ)Z 2 ) ≤ λρ(Z 1 )+(1-λ)ρ(Z 2 ). If the value at risk (VaR), i.e. a quantile of a random variable, is a very popular measure of risk in finance, it appears that it is not a coherent risk measure since it does not satisfy the diversification principle. Oppositely, [START_REF] Artzner | Thinking coherently[END_REF] introduced a Conditional Value at Risk (CVaR) measure (or expected shortfall), which is shown to be a coherent risk measure in [START_REF] Pflug | Some remarks on the value-at-risk and the conditional value-at-risk[END_REF]. More precisely, if X refers to the outcome of a portfolio, we define by F the cumulative distribution function:

∀x ∈ R F (x) = P(X ≤ x),
and for a given α ∈ (0, 1), the quantile θ α is denoted by:

(1)

θ α := inf {θ ∈ R | F (θ) ≥ α} .
Note that in many works in actuarial and financial mathematics, the notation VaR α (X) or ES α (X) (for expected shortfall) is used instead of θ α . For a given α ∈ (0, 1), the superquantile is then defined as soon as the random variable X is L 1 , with the help of:

(2)

ϑ α := E[X | X ≥ θ α ] = E[X1 X≥θα ] 1 -α .
Superquantiles are often adopted as a measure of risk when modeling a risk-averse decision maker and are commonly denoted by CVaR α (X) in financial economics and mathematics. More precisely, ϑ α quantifies a risk-averse constraint on asset return prices: for a (log)-return weekly price Z of a portfolio, we are commonly interested in the estimation or in some guarantees of the super-quantile of -Z when α = 95%, which translates the average loss induced by the worst 5% scenario. Besides being a coherent risk measure, the superquantile also translates more information on the tail of the distribution of the random variable X.

In our paper, we develop a new method for jointly estimating (θ α , ϑ α ) and obtain some non-asymptotic guarantees for the quadratic loss of estimation, which is a novel type of result for such a kind of estimation problem with stochastic algorithms. We also obtain a central limit theorem associated to our method.

Related works. Estimating quantiles has a longstanding history in statistics: except in parametric models where explicit formulas are available, the estimation of quantiles is a real issue. It may be tackled with the help of Monte-Carlo approaches when some simulation tools of the random variable X are available, or using the Extreme Value Theory when α is close to 0 or 1 (see e.g. [DHF06, Emb99]). Another popular point of view is to approximate the behaviour of the VaR α (X) with correcting linear and quadratic terms (see e.g. [BJS99, DP01, Rou97]). Finally, a large amount of work is based on order statistics to derive some estimation procedures of VaR α (X). We refer to [START_REF] Torossian | A review on quantile regression for stochastic computer experiments[END_REF] for an overview of some recent uses of order statistics for quantile estimation with some prospects in the field of computer experiments.

The development of new mathematical studies on superquantiles started with [BTT86] and [RU00], and use some variational formulation of CVaR α (X) as a solution of a convex optimization problem, to derive some estimation strategy. In particular, their approach does not require any parametric form of the distribution of X. In [START_REF] Labopin-Richard | Bregman superquantiles. Estimation methods and applications[END_REF], the authors exploit some Monte-Carlo strategy coupled with some order-statistics to estimate superquantiles and they establish a central limit theorem associated with their estimation method. Oppositely, many (econometric or finance) works are developed with an important parametric a priori modeling (see e.g. [HY03, WZ16, ENW09]) and owing to their parametric or semi-parametric point of view, the methods cited above obviously suffer from robustness issues.

All the previous methods are based on a batch framework, where the user observes the whole set of samples before starting the estimation procedure. Nevertheless, in some concrete situations, the observations may only record on-line, while in other big-data situations, the observations are simply too numerous to be handled with a batch method. In these cases, we are led to consider some recursive strategies where we estimate quantiles and superquantiles using the observations sequentially, with the help of a stochastic approximation algorithm (see, e.g., [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]BMP90]). The recursive quantile algorithm is a classical example of such a method (see e.g. [Duf97]) and has led to recent numerous contributions to the generalization of geometric medians (see, among others [God15, [START_REF] Cardot | Efficient and fast estimation of the geometric median in hilbert spaces with an averaged stochastic gradient algorithm[END_REF]CCGB17]).

The classical algorithm can be written as:

(3)

     θ n+1 = θ n -a n 1 X n+1 ≤θn -α ϑ n+1 = ϑ n + b n X n+1 1 -α 1 X n+1 >θn -ϑ n
Finally, the recent work of [BFP09] developed a different stochastic algorithm that estimates both CVaR α (X) and VaR α (X) within the same joint procedure. Indeed, [BFP09] used the current value of (θ n ) n≥1 and modified (ϑ n ) n≥1 into:

ϑ n+1 = ϑ n + b n θ n + (X n+1 -θ n ) (1 -α) 1 X n+1 >θn -ϑ n = b n (L(θ n , X n+1 ) -ϑ n ),
to estimate ϑ α . Their main idea is to consider a convexified version of the algorithm since the update function

L(θ, x) = θ + x-θ 1-α 1 x>θ satisfies that θ → E(L(θ, X)) is convex.
Then they use a Ruppert-Polyak averaging procedure, introduced in the seminal contributions [Rup88, PJ92], to obtain a central limit theorem for their stochastic algorithm. In the recent work [BCG20], we developed a purely asymptotic study of these two algorithms and derive almost sure properties (Quadratic Strong Law and Law of the Iterated Logarithm) as well as a central limit theorem. 1.2. Algorithm. In this paper, we develop a stochastic procedure, that may be closely related to the one of [BFP09, BCG20], to jointly estimate the quantile and the superquantile of an unknown distribution: we expect to benefit from the acceleration of the Cesaro averaging procedure with ( θn ) n≥1 to obtain a better recursion on the superquantile estimation. For this purpose, we first introduce the following three-dimensional stochastic algorithm, which is a generalization of the stochastic algorithm (3):

(4)                θ n+1 = θ n -a n 1 X n+1 ≤θn -α θn+1 = θn n n + 1 + 1 n + 1 θ n+1 = 1 n n k=1 θ k ϑ n+1 = ϑ n + b n X n+1 1 -α 1 X n+1 > θn -ϑ n .
The sequence (θ n ) n≥1 corresponds to the standard recursive quantile algorithm (see e.g. [Duf97]), whereas ( θn ) n≥1 is the Cesaro average over the past iterations of the initial stochastic gradient descent (θ n ) n≥1 . Cesaro averaging is known as an efficient way for reducing the quadratic risk of estimation of θ α (see e.g. [Rup88, PJ92]). We will provide a sharp nonasymptotic analysis of (θ n ) n≥1 and ( θn ) n≥1 below since it will be necessary for our study of ( ϑ n ) n≥1 . Our approach is similar to [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF] even though for our purpose, we need to derive some upper bounds of the L p risk of ( θn ) n≥1 for p = 4 instead of p = 2 in [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF]. We compare the algorithm proposed in this article with existing ones at the end of this section.

We do not assume any parametric form of this underlying distribution, we only assume that this distribution is absolutely continuous with respect to the real one-dimensional Lebesgue measure λ, whose density is denoted by f :

∀x ∈ R f (x) := dP dλ (x).
We consider the general situation where f is supported over R, i.e. we do not make any assumption on the compactness of the support of f and we do not assume that f is lower bounded by any non-negative constant.

Assumption H f : We assume that f (θ α ) > 0, that f ∞ < +∞ and that θ -→ (1+|θ|)|f (θ)| is a bounded function. We further assume that X has a moment of order strictly larger than 2:

∃p > 2 :

x p f (x)dx < ∞.
In particular, H f implies that f is L-Lipschitz for a large enough L.

Assumption H (an,bn) The gain sequences satisfy:

a n = a 1 n -a and b n = b 1 (n + 1) -b with 1 2 < a < b ≤ 1.
1.3. Main results. We derive non asymptotic estimates of the quadratic loss as well a L p loss of the recursive quantile estimation ( θn ) n≥1 , with p ≥ 1. We introduce the L p risk associated to θn -θ α denoted by M n,2p :

M n,2p = E| θn -θ α | 2p .
Theorem 1.1. Assume H f and H (an,bn) , then: i) A constant Γ a exists such that the recursive quantile algorithm satisfies:

E( θn -θ α ) 2 ≤ α(1 -α) f (θ α ) 2 n + Γ a n -(1+ a 2 )∧( 3 2 -a 2 ) , ∀n ≥ 1.
The optimal choice of a corresponds to a = 2/3 and in this case

E( θn -θ α ) 2 ≤ α(1 -α) f (θ α ) 2 n + Γ 1/2 n -7/6 , ∀n ≥ 1.
ii) For any integer p ≥ 1, a constant c p exists such that:

∀n ≥ 1 M n,2p ≤ c p n -p .
Even though the second result ii) is stated for a general value of p, we emphasize that this result is weaker than that of i) when p = 2: the first item provides a first order optimal result, while the constant c p given by the second item is pessimistic. The first order term is optimal thanks to the Ruppert-Polyak central limit theorem satisfied by ( θn -θ α )

√ n (see e.g. [PJ92, Rup88]). Nevertheless, ii) of Theorem 1.1 will be essential to derive a satisfactory linearization of our superquantile algorithm ( ϑ n ) n≥1 . We also state in Theorem 2.2 similar results for the Robbins Monro algorithme (θ n ) n≥1 , however in this setting we have no explicit expression of the first order constant.

Below, we study the properties of the superquantile recursive estimation algorithm and we differentiate our results into two cases that depend on the choice of the sequence (b n ) n≥0 . We first consider the case b n = b 1 /(n + 1) and then b n = b 1 (n + 1) -b with 1/2 < b < 1. We introduce the important notation V α and prove the following result:

(5)

V α := V (X1 X>θα ) = +∞ θα x 2 f (x)dx - +∞ θα xf (x)dx 2 . Theorem 1.2. Assume H f and H (an,bn) . i) If b ∈ (1/2, 1), then a large enough constant Γ exists such that ∀n ≥ 1 , E(( ϑ n -ϑ α ) 2 ) ≤ V α 2(1 -α) 2 b n + Γn -b+1 2 . ii) If b = 1 and b 1 > ( 1+a 2 ) ∧ ( 5 2 -a)
, then a large enough constant Γ exists such that:

∀n ≥ 1 , E(( ϑ n -ϑ α ) 2 ) ≤ C α,b 1 n + Γn -(1+ a 2 )∧(2-a) ,
where

C α,b 1 = 4b 2 1 α(1 -α) (2b 1 -1) 2 f (θ α ) 2 1 + 1 + V α f (θ α ) 2 (2b 1 -1) 4α(1 -α) 3 2 .
Theorem 1.2 provides some statistical guarantees on the behaviour of ( ϑ n ) n≥1 in a finite horizon n while our assumptions on f are very weak: we only assume the smoothness of f with f (θ α ) > 0. The second assumption seems necessary otherwise the distribution P is flat around θ α and the definition of the quantile and of the super-quantile may be ambiguous.

We also state a sharp asymptotic analysis of the sequence ( ϑ n ) n≥1 and we derive a central limit theorem with an explicit computation of the limiting variance.

Theorem 1.3. Assume H f and H (an,bn) . i) If b ∈ (1/2, 1), then: b -1 n ϑ n -ϑ α L =⇒ n→+∞ N 0, V α 2(1 -α) 2 .
ii) If b = 1, and b 1 > 1/2 then:

√ n θn -θ α ϑ n -ϑ α L =⇒ n→+∞ N (0, S 2 )
where:

S 2 = α(1-α) f (θα) 2 α f (θα) (ϑ α -θ α ) α f (θα) (ϑ α -θ α ) b 2 1 (2b 1 -1) Vα (1-α) 2 -2b 1 2b 1 -1 αθα(ϑα-θα) (1-α)
.

Discussion. We complete below our previous discussion regarding the results we obtained in Theorem 1.2 and Theorem 1.3. Low computational cost.

It is well known that on-line methods lead to very simple and fast ways to compute estimators comparing for example to estimators based on order statistics or plug-in. Hence, it is somewhat classical but however important, to remind how fast our algorithm is to manage the estimation of ϑ α recursively. Moreover, our method may be adapted to a stream of observations, contrary to batch methods. Non-asymptotic result.

Gathering i) of Theorem 1.2 and i) of Theorem 1.3, we observe that our upper bound i) cannot be improved as it involves the lowest possible constant. Regarding now ii) of Theorem 1.2 and ii) of Theorem 1.3, we observe that the rate of convergence in Theorem 1.2 ii) is of the order 1/n and is optimal. Hence, we obtain with our sequential method a parametric rate of estimation of ϑ α . Unfortunately, we emphasize that C α,b 1 given in ii) of Theorem 1.2 does not match with the limiting variance obtained in Theorem 1.3 so that the constant involved in ii) of Theorem 1.2 may certainly be improved. Comparing the results i) and ii) of Theorems 1.2 and 1.3, we finally observe that from an asymptotic point of view, the choice b n = b 1 /(n + 1) always outperform the other one.

To the best of our knowledge, estimating the super-quantile ϑ α with a recursive method has only been addressed by [BFP09] with the help of a Cesaro averaging procedure on the initial sequence (ϑ n ) n≥1 and in [BCG20] where some asymptotic results are established for another method that produces a sequence different from ( ϑ n ) n≥1 . The results in [BFP09] and in [BCG20] are somewhat weaker than that of Theorem 1.2 since with the same set of assumptions, they only derive asymptotic results instead of a non-asymptotic bound on the mean square error. Our contribution stated in Theorem 1.2 is significantly stronger: we obtain a non-asymptotic upper bound of the performance of our estimator, which possesses the good convergence rate O(b n ) with a sequential strategy.

Limiting variance and comparison with existing results

Regarding now the asymptotic properties, there exists roughly speaking three axes of work in the literature. The first family of work is concerned by parametric or semi-parametric approaches: some parametric models are considered for the sequence of observations and then a plug-in parametric estimation is used with either a direct formula (if available for ϑ α ) or a Monte-Carlo integral approximation. It is the point of view adopted in [START_REF] Chernozhukov | Conditional value-at-risk: Aspects of modeling and estimation[END_REF], [START_REF] Hall | Inference in ARCH and GARCH models with heavy-tailed errors[END_REF] or [WZ16] for example. Although efficient from a numerical point of view, these approaches suffer from robustness problems essentially due to some questionable parametrization. Consequently, even if some of these previous works also establish central limit theorem with explicit limiting variance of estimation, they cannot be directly compared to our result stated in Theorem 1.3.

Oppositely, nonparametric (or historical) VaR α (X) or CVaR α (X) estimation is a robust alternative over the (semi-)parametric approaches and among these works, we can mention batch methods and on-line ones that can manage a stream of observations.

• Concerning the batch approaches, they are commonly based either on order statistics, kernel smoothing or on the variational description of CVaR α (X). For example, in [START_REF] Labopin-Richard | Bregman superquantiles. Estimation methods and applications[END_REF], the authors derive a central limit theorem for the estimation of the super-quantile using the standard approach with order statistics and an assumption on the behaviour of the quantile function that entails an assumption on the tail of the distribution of the random variable X, which yields a non-adaptive method regarding the tail parameter of the distribution. In [BZ10], the authors use a plug-in method to derive a central limit theorem, with the help of an ad hoc functional delta method. In the same way, [CU01] also uses a variational approach of CVaR α (X) to derive an asymptotic functional central limit theorem for the estimation of ϑ α whose limiting variance depends on the tail index used in their model. In all these works, the variances are difficult to compare with the one stated in Theorem 1.3, at least from a theoretical point of view, while we emphasize that these methods are computationally longer than our on-line method. We also point out that in our work, we do not need any very restrictive assumption on the tail of the distribution of X.

• Therefore, it seems that the meaningful methods we have to compare with are those of [BCG20], [BFP09] and [START_REF] Labopin-Richard | Bregman superquantiles. Estimation methods and applications[END_REF]. In [START_REF] Labopin-Richard | Bregman superquantiles. Estimation methods and applications[END_REF], the superquantile is estimated using a Monte-Carlo method and some order statistics of the distribution. The authors obtain a central limit theorem (Theorem 3.1) and the limiting variance whose expression is difficult to compare with our because of the lack of explicit computations. More interestingly, [BCG20] states a central limit theorem for the algorithms (ϑ n ) and ( ϑ n ) (see Theorem 3.4). We obtain that both

√ n b (ϑ n -ϑ α ) and √ n b ( ϑ n -ϑ α ) converge to a Gaussian random variable with variance Γ ϑα =        b 2 1 τ 2 α 2b 1 -1 if b = 1, b 1 τ 2 α 2 if b < 1, where τ 2 α = V α (1 -α) 2 - αθ α 1 -α (2ϑ α -θ α ).
In the case where b < 1 and ϑ α and θ α are positive, the asymptotic variance obtained in Theorem 1.3 ii) is larger than Γ ϑα .

The discussion in the case where b = 1 is more interesting. For a fixed value of b 1 we observe that

S 2 22 < Γ ϑα ⇐⇒ b 1 < 2ϑ α -2θ α 2ϑ α -θ α = 1 - θ α 2ϑ α -θ α .
Therefore if the distribution f satisfies that ϑα θα > 3 2 , there exists b 1 such that the asymptotic variance of ϑ n is smaller that the one of ϑ n and ϑ n . Now for a given distribution, the question remains how to find a value of b 1 that minimizes S 2 22 and Γ ϑα . The minimal Γ ϑα is equal to τ 2 α and is reached for b 1 = 1. This value corresponds to the limiting variance of the central limit theorem for the averaged version of ϑ n stated in [BFP09] (Theorem 2.4). The variance S 2 22 admits a minimum for some b 1 ∈ (1/2, 1). However the comparison of this minimal value with τ 2 α is difficult to handle for a generic distribution. We nevertheless point out that our new algorithm ( ϑ n ) n≥1 shall improve the limiting variance of estimation in comparison with the one of (ϑ n ) n≥1 or ( ϑ n ) n≥1 when ϑ α is significantly larger than θ α , which is a common feature with heavy-tail distributions like Student, Weibull or Pareto ones. This last remark may be in particular useful for possible applications in finance and actuary. In these both fields, CVaR estimation is at the cornerstone for designing optimal policies either for contracting or edging. We emphasize that in finance, it is commonly admitted that asset return prices may possess heavy tails (see e.g. [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF][START_REF] Rachev | Stable Paretian Models in Finance[END_REF]) whereas actuarial scientists generally face optimal reinsurance problems with fat tails (see e.g. [BBH09, Del09]).

Organisation of the paper and notations. As indicated above, the main goal of this paper is to prove Theorem 1.2 and Theorem 1.3. To do so, we will also establish some results on the stochastic gradient descent (θ n ) n≥1 involved in the quantile estimation, and on the averaged sequence ( θn ) n≥1 for the quantile estimation itself. Despite their own interests, these results are necessary to study ( ϑ n ) n≥1 and to the best of our knowledge, are new for the recursive quantile algorithm (θ n ) n≥1 and in particular give a state-of-the art result for the sequence ( ϑ n ) n≥1 when compared to known results stated in [Duf97, CCZ13, God15] (among others).

The paper is organized as follows. Section 2 presents a careful study of the L p loss of the recursive quantile. The results derive from a strategy based on a family of Lyapunov functions V q (see Equation 9). In particular, we establish Theorem 2.2 that states that E[(θ n -θ α ) 2p ] is less than a p n up to a constant that depends on p. In Section 2.2, we prove an optimal non-asymptotic result for ( θn ) n≥1 . Our proof relies on a spectral analysis of the algorithm, close to the strategy developed in [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF]. Section 3 then uses Theorem 1.1 to derive the main result of the paper on the super-quantile estimation proof of Theorem 1.2 and Section 4 describes the central limit theorem with the help of an approximation of the rescaled stochastic algorithm by Markov diffusion processes.

Below, we will use a.s. to refer to an almost sure convergence result. For two positive sequences (t n ) n≥1 and (s n ) n≥1 , t n s n refers to an inequality true for any value of n up to a constant C independent of n: t n ≤ Cs n . In the sequel we will always denote (VaR α (X), CVaR α (X)) by (θ α , ϑ α ).

Quantile estimation

This paragraph deals with the statistical estimation of θ α with the sequence (θ n ) n≥1 and the Cesaro averaging procedure defined by ( θn ) n≥1 .

2.1. Convergence of the quantile algorithm (θ n ) n≥1 . We first state a standard almost sure convergence result, established with the help of the Robbins-Monro Theorem.

Theorem 2.1 (Robbins-Monro Theorem -Almost sure convergence of (θ n ) n≥1 ). Assume that a n = a 1 n -a with a ∈ [1/2, 1] and a 1 > 0, then the sequence (θ n ) n≥1 satisfies θ n n-→+∞ -----→ θ α a.s.
We refer to [Duf97] for a proof of this standard result. The major drawback of Theorem 2.1 is the lack of quantitative information about the rate of convergence. We are generally interested in a result of the form

E[(θ n -θ α ) 2q ] ≤ r n,q
, where r n,q is referred to as the convergence rate of the algorithm. We emphasize that these bounds are more meaningful than a simpler almost sure convergence result because they make it possible to draw some quantitative comparisons among algorithms (in particular when looking at the rates of convergence and at the first order terms). Deriving a such bound is at the cornerstone of our forthcoming study of ( θn ) n≥1 .

Theorem 2.2. Assume H f , if a n = a 1 n -a with a ∈ (0, 1), then a large enough constant C exists such that:

M n,4 := E|θ n -θ α | 4 ≤ Cn -2a .
More generally, for any integer q ≥ 1, we have

∃A q ≥ 0 ∀n ∈ N M n,2q = E|θ n -θ α | 2q ≤ A q a q n .
We emphasize that our result holds for a ∈ (0, 1), instead of the classical assumption a ∈ (1/2, 1) in the Robbins-Monro Theorem (see for example the classical result on the dosage in Theorem 1.4.26 and Proposition 1.4.27 of [Duf97]). We prove here that the common constraint n≥1 a 2 n < +∞ is useless in the case of the recursive quantile algorithm to obtain a convergence result. This important and unusual result holds because of the boundedness of the noise from one iteration to another and is obtained with a subtle strong-convexification of the loss function with the help of the family of functions V q . To the best of our knowledge, such improvement has only been observed in the case of bounded noise in a general result of Métivier and Priouret (see e.g. Theorem 1 of [MP87]), and was also stated in a simpler way in Proposition 4.2 of [START_REF] Benaïm | Dynamics of stochastic approximation algorithms[END_REF]. We emphasize that these works only state a.s. asymptotic results. Said differently, for the dosage issue and other companion problems (such as the mediane estimation for example), the assumption a > 1/2 used in [Duf97, God15] to assess convergence results on (θ n ) n≥1 is useless.

Proof of Theorem 2.2. We start with the linearization of (θ n ) n≥1 . We define

(6) Φ(θ) := θ θα u θα f (s)dsdu.
The function Φ is associated to the gradient descent involved by Equation (4). We point out that Φ ≥ 0 and satisfies:

(7) ∀θ ∈ R 0 ≤ Φ(θ) ≤ f ∞ 2 (θ -θ α ) 2 .
We shall remark that:

θ n+1 = θ n -a n (F (θ n ) -F (θ α )) + a n ∆M n+1 = θ n -a n Φ (θ n ) + a n ∆M n+1 ,
where the martingale increment is defined in by: (8)

∆M n+1 := F (θ n ) -1 X n+1 ≤θn .
We now follow the roadmap of [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF] and define the Lyapunov function:

(9) V q (θ) := Φ(θ) q exp(Φ(θ)), ∀q ∈ N.
We first state some important properties of V q , whose proofs are postponed to Appendix B.

Lemma 2.3. i) A constant m > 0 exists such that for any θ ∈ R and q ≥ 1 :

Φ (θ)V q (θ) ≥ mV q (θ).
ii) A constant c q > 0 that only depends on q and f exists such that:

|V q (θ)| ≤ c q (V q (θ) + V q-1 (θ)).
iii) For any q ≥ 1, a positive constant C q exists such that for any θ ∈ R

(θ -θ α ) 2q ≤ C q (V q (θ) + V 2q (θ)) .
Our strategy of proof relies on a recursive contraction from V q (θ n ) to V q (θ n+1 ) using a Taylor expansion of V q (θ n+1 ):

V q (θ n+1 ) = V q θ n -a n Φ (θ n ) + a n ∆M n+1 = V q (θ n ) -a n Φ (θ n ) -∆M n+1 V q (θ n ) + a 2 n (Φ (θ n ) -∆M n+1 ) 2 2 V q (ξ n+1 ),
where (10)

ξ n+1 = θ n + n+1 a n -Φ (θ n ) + ∆M n+1 , with n+1 is a random variable in [0, 1].
We first consider the drift term. Since ∆M n+1 is a martingale increment:

E a n (Φ (θ n ) -∆M n+1 )V q (θ n ) |F n = a n V q (θ n )Φ (θ n ),
and Lemma 2.3 i) yields

E a n (Φ (θ n ) -∆M n+1 )V q (θ n ) |F n > ma n V q (θ n ). (11)
We now pay a specific attention to the second order term. Lemma 2.3 ii) yields:

|V q (ξ n+1 )| ≤ c q (V q (ξ n+1 ) + V q-1 (ξ n+1 )). ( 12
)
We are thus led to upper bound Φ(ξ n+1 ). Using a Taylor expansion near θ n , we obtain:

Φ(ξ n+1 ) = Φ θ n + n+1 a n ∆M n+1 -Φ (θ n ) ≤ Φ(θ n ) + Φ (θ n ) n+1 a n ∆M n+1 -Φ (θ n ) + 2 n+1 a 2 n (∆M n+1 -Φ (θ n )) 2 2 Φ ∞ .
We recall that n+1 is a random variable in [0, 1] and that Φ (θ n ) and ∆M n+1 are uniformly bounded by 1. We thus deduce that:

Φ(ξ n+1 ) ≤ Φ(θ n ) + 2a n + 2a 2 n f ∞ ,
which leads to, for a n large enough:

Φ(ξ n+1 ) ≤ Φ(θ n ) + 3a n .
Changing the constant c q from line to line, we are led to:

V q (ξ n+1 ) ≤ exp(Φ(θ n ))e 3an (Φ(θ n ) + 3a n ) q ≤ c q (V q (θ n ) + a q n V 0 (θ n ))) ≤ c q (V q (θ n ) + a q n (1 + V q (θ n ))) ≤ c q (V q (θ n ) + a q n ) , (13)
where we used that exp(Φ(θ

n )) = V 0 (θ n ) ≤ C(1 + V q (θ n )).
Combining Equations ( 12) and (13) we deduce that:

V q (ξ n+1 ) ≤ c q V q (θ n ) + V q-1 (θ n ) + a q-1 n .
Finally using again the fact that Φ (θ n ) and ∆M n+1 are uniformly bounded, we obtain:

E a 2 n (Φ (θ n ) -∆M n+1 ) 2 2 V q (ξ n+1 ) |F n ≤ c q a 2 n V q (θ n ) + V q-1 (θ n ) + a q-1 n .
We conclude using (11) that:

E[V q (θ n+1 )] = E[E[V q (θ n+1 ) |F n ]] ≤ E (V q (θ n )) 1 -a n m + c q a 2 n + c q a 2 n E (V q-1 (θ n )) + c q a q+1 n . ( 14
)
We deduce from the study of this recursive inequality given in Lemma B.1 that

(15) ∀q ≥ 1, ∃ A q ≥ 0, ∀n ∈ N, E[V q (θ n )] ≤ A q {a n } q .
From Lemma 2.3iii) we have:

(θ -θ α ) 2q ≤ C q (V q (θ) + V 2q (θ)) .
Taking the expectation at time n and using Equation (15), we conclude that:

∃A q ≥ 0 ∀n ∈ N M n,2q ≤ A q a q n .
2.2. Cesaro averaging. This section is devoted to the proof of Theorem 1.1 . The proof relies on a spectral analysis of the algorithm, close to the strategy developed in [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF]. We introduce Z n = (θ n -θ α , θn -θ α ), which gives the joint evolution of the Robbins Monro algorithm and its averaged version.

Proposition 2.4. The sequence (Z n = (θ n -θ α , θn -θ α )) n≥0 satisfies:

(16) Z n+1 = A n Z n + a n ∆M n+1 1 n+1 ∆M n+1 + a n R n 1 n+1 R n
, where: i) A n translates the linearization of the algorithm around (θ α , θ α ) at step n:

A n = 1 -a n f (θ α ) 0 1-anf (θα) n+1 1 -1 n+1 .
ii) The martingale increment ∆M n+1 defined in (8) satisfies:

E {∆M n+1 } 2 |F n -α(1 -α) ≤ f ∞ |θ α -θ n |.
iii) The rest term R n is an F n measurable random variable such that:

(17) |R n | ≤ L 2 |θ n -θ α | 2 .
Proof. Proof of i): We write the two sequences (θ n ) n≥1 and ( θn ) n≥1 as

(θ n+1 -θ α ) = (θ n -θ α ) -a n [F (θ n ) -F (θ α )] + a n ∆M n+1 , and
θn+1 = n n + 1 θn + 1 n + 1 θ n+1 = 1 - 1 n + 1 θn + 1 n + 1 θ n -a n (F (θ n ) -F (θ α )) + a n ∆M n+1 .
We then use a linear approximation of F around θ α and write that:

F (θ n ) -F (θ α ) = f (θ α )(θ n -θ α ) + θn θα [f (u) -f (θ α )]du.
Finally, we can write the joint evolution of (θ n , θn ) n≥1 as a perturbed linear recursive sequence given by ( 16).

Proof of ii):

We study the variance of the martingale increment defined in (8):

E {∆M n+1 } 2 |F n = F (θ n )(1 -F (θ n )) = α(1 -α) + θn θα f (u)[1 -2F (u)]du.
We then deduce that:

E {∆M n+1 } 2 |F n -α(1 -α) ≤ f ∞ |θ α -θ n |.
Proof of iii): The rest term is an F n -measurable random variable given by

R n = θn θα [f (u) -f (θ α )]du.
Using that f is L-Lipschitz, the rest term R n satisfies:

|R n | ≤ L 2 |θ n -θ α | 2 .
Proof of Theorem 1.1 i). The scheme of the proof is to use the linearization to identify the negligible terms in the recursion. To do so, we diagonalize the matrix A n and perform a change of base. The eigenvalues of A n are 1 -a n f (θ α ) and 1 -1 n+1 and we verify that:

A n = 1 0 n 1 1 -a n f (θ α ) 0 0 1 -1 n+1 1 0 -n 1 with n := 1 -a n f (θ α ) 1 -a n (n + 1)f (θ α )
.

We introduce

(18) Z n = 1 0 -n 1 Z n = θ n -θ α -n (θ n -θ α ) + ( θn -θ α ), ,
and we denote by Z

(2)

n the second component of Z n . Equation (16) yields:

(19) Z

(2)

n+1 = 1 - 1 n + 1 Z (2) n + a n ∆M n+1 1 n + 1 -n+1 + R n ,
where:

(20) R n = ω n (θ n -θ α ) + 1 n + 1 -n+1 a n R n with ω n := ( n+1 -n )(1 -a n f (θ α )).
We now compute the quadratic expansion using the fact that (∆M n+1 ) n≥1 is a sequence of martingale increments:

E({ Z (2) n+1 } 2 ) = 1 - 1 n + 1 2 E({ Z (2) n } 2 ) + a 2 n n+1 - 1 n + 1 2 E({∆M n+1 } 2 ) + E(R 2 n ) + 2 1 - 1 n + 1 E(R n Z (2) n ).
We denote

u n = E({ Z (2)
n } 2 ) and use Cauchy-Schwarz inequality on the last term to deduce that:

u n+1 ≤ 1 - 1 n + 1 2 u n + a 2 n n+1 - 1 n + 1 2 E({∆M n+1 } 2 ) + E(R 2 n ) + 2 1 - 1 n + 1 u n E(R 2 n ).
We study in Lemma 2.5 and 2.6 the behaviour of all the terms involved in the previous decomposition. Lemma 2.5 concerns the variance of the martingale increments from which a term of order n -2 arises, while Lemma 2.6 proves that the other terms are negligible.

Lemma 2.5. There exists a constant C (independent from n) such that for all n:

a 2 n n+1 - 1 n + 1 2 E({∆M n+1 } 2 ) ≤ α(1 -α) (n + 1) 2 f (θ α ) 2 + C n -(2+a/2) ∨ n -(3-a) .
Lemma 2.6. There exists a constant C (independent from n) such that for all n:

E(R 2 n ) ≤ C n -(4-a) ∨ n -(2+2a) .
The proof of these results is postponed in Appendix B.2. We deduce from these lemmas that some constant Γ exists such that: 1+a) .

u n+1 ≤ 1 - 1 n + 1 2 u n + α(1 -α) (n + 1) 2 f (θ α ) 2 + Γ n -(2+a/2) ∨ n -(3-a) + Γ √ u n n -(2-a/2) ∨ n -(
We now apply Lemma A.2 in order to derive the convergence rate of the algorithm. With the notations introduced in Lemma A.2 applied with γ = 1, A 1 = 0 and:

C = α(1 -α) f (θ α ) 2 and r 2 = (2 + a 2 ) ∧ (3 -a) and r 1 = (2 - a 2 ) ∧ (1 + a).
We then obtain that a large enough Γ a exists such that:

u n ≤ α(1 -α) f (θ α ) 2 n + Γ a n -r ,
where r is given by:

r = 1 + a 2 ∧ (2 -a) ∧ 3 2 - a 2 ∧ 1 2 + a = 3 2 - a 2 ∧ 1 2 + a , as a ∈ [1/2, 1
). The optimal value of r is attained for a = 2/3 and the global rate is then:

(21) E({ Z (2) n } 2 ) ≤ α(1 -α) f (θ α ) 2 n + Γ 2/3 n -7/6 .
To conclude the proof, we come back to E({ θn -θ α } 2 ). From the definition of Z

(2)

n we obtain that:

E({ θn -θ α } 2 ) = E{ Z (2) n } 2 + 2 n E( Z (2) n (θ n -θ α )) + 2 n E((θ n -θ α ) 2
). Therefore, it remains to prove that the two last terms are negligible with respect to n -r . From Theorem 2.2, we know that a large enough C exists such that

2 n E((θ n -θ α ) 2 ) ≤ C n 2 a n ,
and the Cauchy-Schwarz inequality yields (up to a modification of C):

n E( Z (2) n (θ n -θ α )) ≤ Cn -3/2-a/2 .
Proof of Theorem 1.1-ii). The proof is deferred to Appendix B.2.

Embedded averaging super-quantile algorithm

The purpose of this paragraph is to prove Theorem 1.2, i.e., we aim to derive a nonasymptotic result on the sequence ( ϑ n ) (see the definition in Equation (4)).

3.1. Linearization of the embedded algorithm. To study the behavior of Algorithm (4), we follow the roadmap of Section 2 and write a linear approximation of ( ϑ n ) n≥1 . To this end, we define:

(22) Q : θ -→ E [X1 X≥θ ] ,
and

(23) ∆N n+1 := X n+1 1 X n+1 ≥ θn -Q( θn ).
We shall prove the following result.

Proposition 3.1. Assume H f , for any integer n, one has

ϑ n+1 -ϑ α = ( ϑ n -ϑ α ) (1 -b n ) - b n 1 -α ( θn -θ α )θ α f (θ α ) + b n 1 -α Q (ξ n ) ( θn -θ α ) 2 2 + b n 1 -α ∆N n+1 , ( 24
)
where ξ n ∈ ( θn , θ α ),V α is defined in (5) and ∆N n+1 verifies:

(25) E[{∆N n+1 } 2 |F n ] -V α ≤ C| θn -θ α | and |E{∆N n+1 } 2 -V α | ≤ C M n,2 .
Proof. We observe that:

ϑ n+1 -ϑ α = ϑ n -ϑ α + b n 1 X n+1 ≥ θn X n+1 1 -α -ϑ n = ϑ n -ϑ α + b n 1 -α X n+1 (1 X n+1 ≥ θn -1 X n+1 ≥θα ) + b n 1 X n+1 ≥θα X n+1 1 -α -ϑ n .
According to Equations ( 22) and (23), we obtain that:

ϑ n+1 -ϑ α = ϑ n -ϑ α (1 -b n ) + b n 1 -α [Q( θn ) -Q(θ α )] + b n 1 -α ∆N n+1 .
We use a second order Taylor expansion on Q near θ α using the relationships:

Q(θ) = +∞ θ xf (x)dx, Q (θ) = -θf (θ) and Q (θ) = -f (θ) -θf (θ).
In particular, H f implies that Q ∞ < +∞. We obtain that:

ϑ n+1 -ϑ α = ( ϑ n -ϑ α ) (1 -b n ) - b n 1 -α ( θn -θ α )θ α f (θ α ) + b n 1 -α Q (ξ n ) ( θn -θ α ) 2 2 + b n 1 -α ∆N n+1 , where ξ n ∈ [θ α , θn ].
Now, we compare the variance of the martingale increment and V α :

E[{∆N n+1 } 2 |F n ] = E X 2 n+1 1 X n+1 ≥ θn |F n -E X n+1 1 X n+1 ≥ θn |F n 2 = ∞ θn x 2 f (x)dx - +∞ θn xf (x)dx 2 = +∞ θα x 2 f (x)dx - +∞ θα xf (x)dx 2 + δ n+1 ,
where:

δ n+1 = θα θn x 2 f (x)dx - +∞ θn xf (x)dx 2 + +∞ θα xf (x)dx 2 = θα θn x 2 f (x)dx + θn θα xf (x)dx 2 +∞ θα xf (x)dx + θα θn xf (x)dx .
Using H f , x → x 2 f (x) is bounded so that a large enough constant C exists such that:

|δ n+1 | ≤ C| θn -θ α |.
Computing the whole expectation and using the Cauchy-Schwarz inequality, we have that:

E{∆N n+1 } 2 -V α ≤ C M n,2 .
3.2. Spectral analysis and recursion. We now study the sequence ( ϑ n ) with the help of the previous linearization combining Propositions 3.1 and 2.4. We introduce the matrix (26)

B n :=    1 -a n f (θ α ) 0 0 1-anf (θα) n+1 1 -1 n+1 0 0 -θαf (θα) 1-α b n 1 -b n    .
We emphasize that the eigenvalues of B n are

Sp(B n ) = 1 -a n f (θ α ); 1 - 1 n + 1 ; 1 -b n .
Obviously, B n may be reduced to a diagonal form as soon as b n / ∈ {a n f (θ α ), 1 n+1 }. In particular, we obtain the following decomposition (stated without any proof). Proposition 3.2. For any integer n, and if b n / ∈ {a n f (θ α ), 1 n+1 }, we have

B n = P n D n P -1 n with D n =   1 -a n f (θ α ) 0 0 0 1 -1 n+1 0 0 0 1 -b n   ,
where P n is the matrix of change of basis:

P n =   1 0 0 n 1 0 n δ n k α,n 1   , P -1 n =   1 0 0 -n 1 0 -n δ n + n k α,n -k α,n 1   , with: n = 1 -a n f (θ α ) 1 -a n (n + 1)f (θ α ) , δ n = θ α f (θ α ) (1 -α) a n b n f (θ α ) -1) -1 , k α,n = θ α f (θ α )b n (n + 1) (1 -α)(1 -b n (n + 1))
.

As pointed out above, B n may not be reduced to a diagonal form for any sequence b n -→ 0. Therefore, instead of using an exact spectral decomposition as it is the case in [START_REF] Gadat | Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity[END_REF], the novelty of our work here is to handle a decomposition of B n without an exact diagonal form. For this purpose, we introduce the invertible matrix P n defined by:

P n =   1 0 0 n 1 0 n δ n 0 1   ,
and we verify that:

(27) P -1 n B n P n = D n -b n θ α f (θ α ) 1 -α   0 0 0 0 0 0 0 1 0   :=E 3,2 = D n -b n θ α f (θ α ) 1 -α E 3,2 .
Remark 1. The matrix P n corresponds to setting k α,n = 0 which removes the singularity issue since k α,n → ∞ for b n close to 1/(n + 1).

Equation ( 27) may be used as follows: we introduce the vector defined by:

O n :=   θ n -θ α θn -θ α ϑ n -ϑ α   ,
and we remark that Proposition 3.1 yields:

O n+1 = B n O n +   a n ∆M n+1 an n+1 ∆M n+1 bn (1-α) ∆N n+1   +   a n R n an n+1 R n b n R n   where R n := Q (ξ n ) (1 -α) ( θn -θ α ) 2 2 .
Using P n , we translate the spectral information on B n to the new vector:

O n := P -1 n O n . Equation (27) yields: O n+1 = P -1 n+1 O n+1 = D n -b n θ α f (θ α ) 1 -α E 3,2 O n + ( P -1 n+1 P n -I 3 ) D n -b n θ α f (θ α ) 1 -α E 3,2 O n + P -1 n+1   a n R n an n+1 R n b n Rn   + P -1 n+1   a n ∆M n+1 an n+1 ∆M n+1 bn 1-α ∆N n+1   .
We define:

(28)

η n := n δ n -n+1 δ n+1 ,
and verify that:

( P -1 n+1 P n -I 3 ) D n -b n θ α f (θ α ) 1 -α E 3,2 =   0 0 0 (1 -a n f (θ α ))( n -n+1 ) 0 0 (1 -a n f (θ α ))η n 0 0   .
Considering the third coordinate of the sequence ( O n ) n≥1 , we obtain the recursion:

O (3) n+1 = (1 -b n ) O (3) n -b n θ α f (θ α ) 1 -α O (2) n + (1 -a n f (θ α ))η n O (1) n -n+1 δ n+1 a n R n + b n R n -n+1 δ n+1 a n ∆M n+1 + b n (1 -α) ∆N n+1 .
All negligible F n -measurable terms are gathered in U n defined by: (29)

U n := -b n θ α f (θ α ) 1 -α O (2) n + (1 -a n f (θ α ))η n O (1) n -n+1 δ n+1 a n R n + b n R n ,
and we regroup the martingale increments into ∆V n+1 defined by:

∆V n+1 = n+1 δ n+1 a n ∆M n+1 + b n (1 -α) ∆N n+1 .
Then the recursion reads:

(30) O

(3)

n+1 = (1 -b n ) O (3) n + U n + ∆V n .
The next result gives a recursive inequality on

W n = E O (3) n 2 .
Proposition 3.3. Assume that H f and H (an,bn) hold, then

W n = E{ O (3)
n } 2 satisfies:

(31)

W n+1 ≤ (1 -b n ) 2 W n + b 2 n V α (1 -α) 2 + C W n n -(2-a/2) ∨ n -(b+1/2) + C n -(2b+a/2) ∨ n -(3-a) ∨ n -(2b+1) ,
where V α is given in (5) and C > 0.

Proof. A direct square expansion of (30) yields:

W n+1 ≤ (1 -b n ) 2 W n + E(U 2 n ) + 2 (1 -b n ) E[ O (3) n U n ] + E(∆V 2 n+1 )
The Cauchy-Schwarz inequality entails 2 (1

-b n ) E[ O (3) n U n ] ≤ 2|1 -b n | E[U 2 n ] √ W n .
The next lemmas derive some bounds on E(U 2 n ) and E(∆V 2 n+1 ). The proofs are given in Appendix C.

Lemma 3.4. Assume that H f and H (an,bn) hold, then: 2b+1) . Moreover, in the special case where b n = b 1 (n + 1) -1 , a positive C exists such that:

E(U 2 n ) n -(
E(U 2 n ) ≤ 4 α(1 -α) f (θ α ) 2 b 2 1 n -3 + Cn -(4-a)∧( 5 2 +a) .
Lemma 3.5. Assume H f and H (an,bn) hold, then:

E(∆V 2 n+1 ) = b 2 n V α (1 -α) 2 + O b 2 n a 1/2 n ∨ n -(3-a) .
We apply Lemma 3.5 and Lemma 3.4 and obtain that a large enough C > 0 exists such that:

W n+1 ≤ (1 -b n ) 2 W n + b 2 n V α (1 -α) 2 + C W n n -(b+1/2) + Cn -(2b+a/2)∧(3-a) .
We conclude this section with the proof of Theorem 1.2.

Proof of Theorem 1.2.

Case b n = b 1 (n + 1) -b with b ∈ (1/2, 1). We apply Lemma A.1 to the recursion obtained in Proposition 3.3, with

C = V α (1 -α) 2 , r 1 = b + 1/2 and r 2 = (2b + a/2) ∧ (3 -a).
Then, we easily verify that the largest possible value of p in Lemma A.1 is

p = r 1 - b 2 ∧ r 2 -b ∧ 2b ∧ 1 = b + 1 2 .
Since b < p, we shall apply Lemma A.1 and we obtain that a large enough Γ a exists such that:

W n ≤ V α 2(1 -α) 2 b n + Γn -b+1 2 .
Case b n = b 1 (n + 1) -1 . In this case we use Lemma 3.4 to refine the recursion obtained in Proposition 3.3 into: a+1) .

W n+1 ≤ (1 - b 1 n + 1 ) 2 W n + b 2 1 V α (1 -α) 2 (n + 1) 2 + C n -(2+a/2)∧(3-a) + 2 W n (1 -b n ) 4α(1 -α) f (θ α ) 2 b 1 n -3/2 (1 -b n ) 1 + Cn -(1-a)∧(a-1/2) ≤ (1 - b 1 n + 1 ) 2 W n + b 2 1 V α (1 -α) 2 (n + 1) 2 + C n -(2+a/2)∧(3-a) + W n 4b 1 α(1 -α) f (θ α ) n -3/2 + Cn -(5/2-a)∧(

We now use Lemma

A.2 with γ = b 1 , C = b 2 1 Vα (1-α) 2 , A 1 = 4b 1 α(1-α) f (θα) 2 , r 1 = (5/2 -a) ∧ (a + 1) and r 2 = (2 + a/2) ∧ (3 -a). If we choose ρ = (1 + a 2 ) ∧ (2 -a) and C α,b 1 = A 1 2(2γ -1) + A 2 1 4(2γ -1) 2 + C 2γ -1 2 = 4b 2 1 α(1 -α) (2b 1 -1) 2 f (θ α ) 2 1 + 1 + V α f (θ α ) 2 (2b 1 -1) 4α(1 -α) 3 2 ,
then we obtain that for any b 1 > ρ -1/2:

W n ≤ C α,b 1 n + Γn -(1+a/2)∧(2-a) ,
and this inequality is made optimal when a = 2 3 , which entails a second order term equals to n -4/3 when b 1 > 5/6.

Conclusion:

To conclude the proof, we come back to E({ ϑ n -ϑ α } 2 ). From the definition of O

(3) n we obtain

E({ ϑ n -ϑ α } 2 ) = E({ O (3) n } 2 ) + 2 n δ n E( O (3) n (θ n -θ α )) + 2 n δ 2 n E((θ n -θ α ) 2
). Therefore, it remains to prove that the two last terms are negligible with respect to the first one. From Theorem 2.2, we know that a large enough C exists such that

2 n δ 2 n E((θ n -θ α ) 2 ) ≤ Cn -2-2b+3a ,
and the Cauchy-Schwarz inequality yields (up to a modification of C):

n δ n E( O (3) n (θ n -θ α )) ≤ Cn -(1+ 3 2 b-3 2 a) .
When b = 1, an easy comparison leads to:

n -4+3a + n -(5/2-3/2a) n -(1+a/2)∧(2-a) -→ 0.
When b ∈ (1/2, 1) and a < b, we can verify that both terms are negligible when compared to n -(1+a)/2 .

Central Limit Theorem

Deriving a central limit theorem for a stochastic algorithm is a common way to assess the asymptotic variance of the algorithm. Central limit theorem for superquantiles has been addressed in [BFP09] for the Cesaro averaging of the sequence (ϑ n ) n≥1 . Another related relevant work on the subject is the paper of [MP06] where a general central limit theorem is obtained for stochastic algorithms with two different time scales that allows to study a weak convergence result for the pair (θ n , ϑ n ) (see also [BCG20]). We also refer to the pioneering works of [START_REF] Borkar | Stochastic approximation with two time scales[END_REF][START_REF] Konda | Convergence rate of linear two-time-scale stochastic approximation[END_REF] for other results on two time-scales algorithms. However, in our case, even though (θ n , ϑ n ) n≥1 seems to evolve with two different time scales (a n , b n ) n≥1 , the evolution of ( ϑ n ) n≥1 from iteration n to iteration n + 1 highly depends on θn and therefore a third time-scale (1/(n + 1)) is involved. This last remark significantly complicates the use of Theorem 1 of [MP06]. All the more, it appears that some of the technical assumptions used in [MP06] (especially Assumption (A4)-iii)) can be avoided with the help of another proof strategy.

In what follows, we will first consider the case where b n = b 1 (n + 1) -1 in which the convergence speed of (ϑ n ) n≥1 is optimal, and derive a joint convergence for ( θn , ϑ n ) that evolves with a single time scale proportionnal to 1/n (see Equation 4). To derive a central limit theorem for our pair ( θn , ϑ n ) n≥1 , we use the diffusion approximation of stochastic algorithm and we follow the roadmap of [BMP90] (see also [START_REF] Kushner | Stochastic approximation and recursive algorithms and applications[END_REF]). We write the joint evolution of a rescaled version of the algorithm ( θn , ϑ n ) n≥1 and introduce:

(32)

T n = √ n( θn -θ α ) and V n = 1 b n ( ϑ n -ϑ α ).
4.1. Case of a step sequence b n = b 1 /(n + 1).

4.1.1. Decomposition of (T n ) n≥1 and (V n ) n≥1 . To obtain a central limit theorem, we will use an approximation of the rescaled algorithm by a stochastic differential equation. More precisely, we will show that the pair (T n , V n ) n≥1 is close to the discretization of a specific Markov process: the Ornstein-Uhlenbeck diffusion. Of course, if one considers the first coordinate, we will then recover the central limit theorem for the Cesaro averaging of the quantile sequence. We emphasize that our method significantly differs from the ones previously developed to establish central limit theorem results for the Ruppert-Polyak averaging algorithm:

[PJ92], [START_REF] Pelletier | Asymptotic almost sure efficiency of averaged stochastic algorithms[END_REF],[CCGB17] are essentially based on the Abel transform and the decomposition of Equation (48).

Recursive analysis of ( Tn ) n≥1 . Instead of directly studying the pair (T n , V n ) n≥1 , we first consider the reduction of ( θn -θ α ) obtained after our eigenvalue decomposition in Section 2.2. We recall the definition of ( Z n ) n≥1 in (18) and we introduce:

Tn := √ n Z (2) n = √ n ( θn -θ α ) -n (θ n -θ α ) = T n - √ n n (θ n -θ α ).
We remark, using (19), that:

Tn+1 = √ n + 1 Z (2) n+1 = √ n + 1 n n + 1 Z (2) n + a n 1 n + 1 -n+1 ∆M n+1 + R n = n n + 1 Tn + a n √ n + 1 1 n + 1 -n+1 ∆M n+1 + √ n + 1R n .
Finally

Tn+1 = Tn - 1 2(n + 1) Tn + a n √ n + 1 1 n + 1 -n+1 ∆M n+1 + Ẽ1 n , (33)
where Ẽ1

n satisfies:

(34) | Ẽ1 n | | Tn | n 2 + |θ n -θ α | n 3/2-a + (θ n -θ α ) 2 √ n .
The previous upper bound is obtained using a Taylor approximation of n/(n + 1) on the first term, the control R n defined in (20) given by Lemma 2.6, the definition of ( n ) n≥1 and Equation ( 17). Recursive analysis of (V n ) n≥1 . We use the recursive formulation given in Proposition 3.1:

V n+1 = V n 1 + 1 n + 1 1 - b 1 n + 1 - √ b 1 θ α f (θ α ) 1 -α √ n + 2 √ n(n + 1) T n + √ b 1 (1 -α) √ n + 2 n + 1 ∆N n+1 + √ b 1 1 -α √ n + 2 n + 1 Q (ξ n+1 )( θn -θ α ) 2 .
Using T n = Tn + n √ n(θ n -θ α ) and a first order Taylor expansion of 1 + 1/(n + 1) and of (n + 2)/n, we deduce that:

V n+1 = V n - b 1 -1 2 n + 1 V n - √ b 1 θαf (θα) 1-α n + 1 Tn + √ b 1 (1 -α) √ n + 2 n + 1 ∆N n+1 + Ẽ2 n , (35)
where Ẽ2

n satisfies: 33) and (35), we have obtained the following proposition.

(36) | Ẽ2 n | |V n | n 2 + | Tn | n 2 + | θn -θ α | 2 √ n + |θ n -θ α | n 3/2-a . Gathering Equations (
Proposition 4.1. For any n ≥ 1, define Zn = ( Tn , V n ), then:

Zn+1 = Zn - 1 n + 1 1/2 0 √ b 1 θαf (θα) 1-α b 1 -1/2 Zn + Ẽn :=Hn( Zn) + 1 √ n + 1   a n (n + 1) 1 n+1 -n+1 ∆M n+1 √ b 1 (1-α) n+2 n+1 ∆N n+1   :=∆M n+1 , where Ẽn = ( Ẽ1 n , Ẽ2 n ) satisfies: Ẽn | θn -θ α | n 3/2 + | θn -θ α | 2 √ n + |θ n -θ α | n 3/2-a + |θ n -θ α | 2 √ n + | ϑ n -ϑ α | n 3/2 . (37)
4.1.2. Tightness and limit of the martingale increments. We use some standard notations of diffusion approximations of rescaled stochastic algorithms. We introduce:

Γ n = n k=1 1 k ,
The number of iterations needed to last a time t after a shift of Γ n is then denoted by t n :

t n = Γ N (n,t) -Γ n and N (n, t) = max m ≥ n, m k=n+1 1 k < t .
We associate to the sequence ( Zn ) n≥1 a sequence ( Z(n) ) n≥1 of time-shifted continuous processes defined as follows: for any integer n, the process ( Z(n) t ) t≥0 corresponds to a continuous-time affine interpolation of the rescaled stochastic algorithm that starts at position Zn at time 0:

Z(n) t = ZN(n,t) + (N (n, t) + 1)(t -t n )( ZN(n,t)+1 -ZN(n,t) ).
Using the notations of Proposition 4.1 we remark that:

(38) ∀t ≥ 0 Z(n) t = Zn + B (n) t + M (n)
t , where:

B (n) t = - N (n,t) k=n+1 H k-1 (Z k-1 ) -(t -t n )(N (n, t) + 1)H N (n,t) (Z N (n,t) ),
and:

M (n) t = N (n,t) k=n+1 1 k ∆M k + N (n, t) + 1(t -t n )∆M N (n,t)+1 .
In this paragraph, we identify the possible weak limits of the sequence Z(n) (see e.g.

[BMP90]) and some details are skipped for the sake of convenience. The reader may found similar arguments in Section 5 of [START_REF] Gadat | Stochastic heavy ball[END_REF].

We define the infinitesimal generator G : for any ϕ ∈ C 2 b (R 2 , R):

G(ϕ)(x, y) = - x 2 ∂ x ϕ - √ b 1 θ α f (θ α ) 1 -α x + (b 1 -1/2)y ∂ y ϕ + α(1 -α) 2f (θ α ) 2 ∂ 2 xx ϕ + b 1 V α 2(1 -α) 2 ∂ 2 yy ϕ + b 1 αϑ α f (θ α ) ∂ 2 xy ϕ. (39) Proposition 4.2.
i) The sequence ( Z(n) ) n≥1 is tight for the weak topology induced by the weak convergence on compact intervals of continuous time processes. ii)

If V α is defined in (5) then E[∆M n+1 ∆M t n+1 |F n ] = α(1-α) f (θα) 2 √ b 1 αϑα f (θα) √ b 1 αϑα f (θα) b 1 Vα (1-α) 2 + υ n , where υ n -→ 0 in L 1 . iii) For any function ϕ ∈ C 2 b (R 2 , R), E ϕ( Zn+1 ) | Zn = ϕ( Zn ) + 1 n + 1 G(ϕ)( Zn ) + Q n ,
where lim n-→+∞ nE|Q n | = 0.

The proof of this previous compactness result is deferred to Appendix D.

4.1.3. Central Limit Theorem -Theorem 1.3-ii).

Proof of Theorem 1.3-ii). The proof is briefly sketched since it exactly follows the same lines as those of Theorem 2.4 of [GPS18] (Section 5.3).

• From Proposition 4.2-i), we know that the sequence of processes ( Z(n) ) n≥1 is tight and Proposition 4.2-ii)/iii) entails that any weak limit of ( Z(n) ) n≥1 is solution of the martingale problem with generator G on the domain of twice differentiable functions C 2 b (R 2 , R). We emphasize that G corresponds to an Ornstein-Uhlenbeck Markov generator. The associated semi-group P t = e tG is elliptic and ergodic: a unique invariant Gaussian distribution ν ∞ exists such that for any test function ϕ: P t (ϕ) -→ ν ∞ (ϕ) as t -→ +∞ uniformly on compact sets of R 2 . We compute the limit covariance in Proposition 4.2-ii).

• Second, the normalized algorithm ( Zn ) n≥1 is also a tight sequence: we consider a possible weak limit µ associated to an extracted sequence ( Zn k ) k≥1 and a time T large enough. Then, we prove that an associated subseqence of shifted processes Z(m k ) exists such that Z(m k )

T is arbitrarily close to ν ∞ and to µ, because of the shifted nature of continuous time processes.

• We then conclude that µ = ν ∞ , meaning that ( Zn ) n≥1 is a tight sequence with a unique adherence point. It proves that Zn L -→ n→+∞ ν ∞ . We recall that ν ∞ is a bi-dimensional gaussian with mean 0 and covariance matrix Σ. We finally obtain the convergence of √ n( θn -θ α , ϑ n -ϑ α ) using the Slutsky theorem. Let us remark that:

√ n θn -θ α ϑ n -ϑ α = Z(1) n + √ n n (θ n -θ α ) √ b 1 n n+1 Z(2) n .
We verify that:

E √ n n |θ n -θ α | 1 √ na n -→ 0 as n -→ +∞,
so that √ n n (θ n -θ α ) converges to 0 in probability (even in L 1 ). We therefore deduce that √ n( θn -θ α , ϑ n -ϑ α ) converges to a Gaussian distribuion with mean 0 and covariance matrix

S 2 = Σ xx √ b 1 Σ xy √ b 1 Σ xy b 1 Σ yy .
• We finally detail the computation of the limiting variance Σ and identify the invariant measure ν ∞ associated to the infinitesimal generator G defined in (39). We apply Gϕ(x, y)ν ∞ (dx, dy) = 0 to ϕ(x, y) = x 2 /2 and observe that:

- x 2 2 + α(1 -α) 2f (θ α ) 2 ν ∞ (
dx, dy) = 0, which leads to:

Σ xx = α(1 -α) f (θ α ) 2 .
Then using the function ϕ(x, y) = xy, we obtain that:

- xy 2 - √ b 1 θ α f (θ α ) 1 -α x + (b 1 -1/2)y x + √ b 1 αϑ α f (θ α ) ν ∞ (dx, dy) = 0.
Therefore, we deduce that:

Σ xy b 1 = √ b 1 αϑ α f (θ α ) - √ b 1 θ α f (θ α ) 1 -α Σ xx ,
and

Σ xy = α f (θ α ) √ b 1 (ϑ α -θ α ).
Finally using ϕ(x, y) = y 2 /2, we have

- √ b 1 θ α f (θ α ) 1 -α x + (b 1 -1/2)y y + b 1 V α 2(1 -α) 2 ν ∞ (dx, dy) = 0,
and thus

Σ yy = 2 2b 1 -1 b 1 V α 2(1 -α) 2 - αθ α (1 -α) (ϑ α -θ α ) . 4.2. Case of a step sequence b n = b 1 /n b .
Since the computations are rather similar, we only sketch the essential steps of the proof.

Proof of Theorem 1.3-i). We still use Proposition 3.1 as:

V n+1 = V n b n b n+1 (1 -b n ) + b n b n b n+1 ∆N n+1 1 -α - b n b n+1 ( θn -θ α ) θ α f (θ α ) 1 -α + b n b n+1 (1 -α) Q (ξ n+1 )( θn -θ α ) 2 .
Following Proposition 4.1, we can identify the main part of the recursion and the rest terms:

V n+1 = V n -V n b n + E n+1 + b n ∆N n+1 1 -α b n b n+1
,

where

E n+1 = V n (1-b n ) b n b n+1 -1 - b n b n+1 ( θn -θ α ) θ α f (θ α ) 1 -α + b n b n+1 (1 -α) Q (ξ n+1 )( θn -θ α ) 2 .
Let us remark that:

|E n+1 | |V n | n + n -b/2 | θn -θ α |+n -b/2 | θn -θ α | 2 .
Then, the arguments developed in the previous sections can be easily adapted. In the following we only detail the modifications and skip below the technical details related to the discrete/continuous time-scale.

Tightness of (V n ) n≥1 . The proof is similar to the one stated in Appendix D. Indeed, sup n≥1 E V n 2 < +∞ and for any δ > 0,

N (n,t+δ)+1 N (n,t) b k ≤ 2δ for large n, thus (40)
P   N (n,t+δ)+1 N (n,t) b k V k ≥   -2 δ 2 .
The only modification comes from the rest terms:

N (n,t+δ)+1 N (n,t) E k 2   N (n,t+δ)+1 N (n,t) | θk -θ α | k b/2 + | θk -θ α | 2 k b/2 + | ϑ k -ϑ α | k 1-b/2   2   N (n,t+δ) N (n,t) | θk -θ α | k b/2   2 +   N (n,t+δ) N (n,t) | θk -θ α | 2 k b/2   2 +   N (n,t+δ) N (n,t) | ϑ k -ϑ α | k 1-b/2   2 .
We apply the Cauchy-Schwarz inequality:

N (n,t+δ)+1 N (n,t) E k 2   N (n,t+δ)+1 N (n,t) b k   ×   N (n,t+δ)+1 N (n,t) | θk -θ α | 2 + | θk -θ α | 4 + | ϑ k -ϑ α | 2 k 2-2b   .
We shall now compute the expectation of the previous terms and verify that:

E N (n,t+δ) N (n,t) Ẽk 2   N (n,t+δ) N (n,t) b k     N (n,t+δ) N (n,t) b k E| θk -θ α | 2 b k + E| θk -θ α | 4 b k + E| ϑ k -ϑ α | 2 k 2-b   .
Using now Theorem 1.1 (L p loss on the sequence ( θn ) n≥1 ) and Theorem 1.2 (L 2 loss on the sequence (ϑ n ) n≥1 ), we obtain that the bracket term is uniformly bounded in k, and therefore:

E N (n,t+δ) N (n,t) Ẽk 2   N (n,t+δ) N (n,t) b k   2 δ 2 .
Hence, the sequence V n is tight. Finally, applying the above arguments and the bound on E n , any weak limit is solution of the martingale problem associated with the generator:

G(ϕ)(x) = -xϕ (x) + V α 2(1 -α) 2 ϕ (x). ( 41
)
It remains to identify the invariant measure associated with the Ornstein-Ulhenbeck diffusion (41). An easy computation with ϕ(x) = x 2 /2 yields the next result.

ϑ n -ϑ α √ b n L -→ n→+∞ N 0, V α 2(1 -α) 2 .
Now remark that Φ given by Φ (θ) = θ θα f (s)ds is decreasing on ] -∞, θ α ] and and then increasing on [θ α , +∞[. In particular, a c ε > 0 exists such that:

∀θ / ∈ [θ α -ε, θ α + ε] Φ 2 (θ) ≥ c ε > 0.
As the sum of the two previous terms, we deduce that a real value m > 0 exists such that

∀θ ∈ R q Φ (θ) 2 Φ(θ) + Φ (θ) 2 ≥ m > 0.
Proof of ii): Since Φ = f and V q = ΦV q-1 , Equation (44) yields

|V q (θ)| ≤ V q-1 (θ)|f | ∞ [q + Φ(θ)] + Φ (θ) 2 V q-1 (θ) q(q -1) Φ(θ) + 2q + Φ(θ) = V q (θ) |f | ∞ + 2q Φ (θ) 2 Φ(θ) + Φ (θ) 2 + V q-1 (θ) q|f | ∞ + q(q -1) Φ (θ) 2 Φ(θ) . We have shown in i) that Φ 2 Φ -1 is bounded near θ α whereas the unique zero of Φ is θ α and Φ satisfies |Φ | ∞ ≤ (1 -α) ∨ α. Consequently, Φ 2 Φ -1 is a bounded function. Finally, a constant c q > 0 exists such that V q (θ) ≤ c q [V q (θ) + V q-1 (θ)]
Proof of iii) It remains to derive a lower bound of Φ(θ) that involves powers of (θ -θ α ).

Similarly to Equation (45), we consider a small ε such that ∀θ ∈

[θ α -ε, θ α + ε], Φ(θ) ≥ f (θα) 2 (θ-θα) 2 2
and Φ (θ) ≥ f (θα) 2 (θ -θ α ). Then we have for θ ≥ θ α :

Φ(θ) = θ θα u θα f (s)dsdu = θ∧(θα+ε) θα u θα f (s)dsdu + θ θ∧(θα+ε) u θα f (s)dsdu
Differentiating whether θ > θ α + ε or not, and using the previous bound we deduce

Φ(θ) ≥ 1 θ≤θα+ε f (θ α ) 2 (θ -θ α ) 2 2 + 1 θ>θα+ε f (θ α ) 2 ε 2 2 + 1 θ>θα+ε θ θα+ε θα+ε θα f (s)d(s)dsdu ≥ f (θ α ) 2 1 θ≤θα+ε (θ -θ α ) 2 2 + 1 θ>θα+ε ε 2 2 + ε(θ -θ α -ε) ≥ f (θ α ) 2 1 θ≤θα+ε (θ -θ α ) 2 2 + 1 θ>θα+ε ε(θ -θ α ) - ε 2 2
Up to choosing a smaller ε we finally deduce that:

Φ(θ) ≥ f (θ α ) 2 1 θ≤θα+ε (θ -θ α ) 2 2 + 1 θ>θα+ε ε(θ -θ α ) 2 .
We repeat the same arguments when θ ≤ θ α and deduce that a large enough constant C q exists such that:

(θ -θ α ) 2q = (θ -θ α ) 2q 1 |θ-θα|≤ε + (θ -θ α ) 2q 1 |θ-θα|>ε ≤ C q Φ(θ) q 1 |θ-θα|≤ε + Φ(θ) 2q 1 |θ-θα|>ε ≤ C q (V q (θ) + V 2q (θ)) .
Lemma B.1. Let us consider the function V q introduced in (9) and recall that there exists n 0 > 0 such that for any q ≥ 1, a c q > 0 exists such that:

∀n ≥ n 0 E[V q (θ n+1 )] ≤ E (V q (θ n )) 1 -a n m + c q a 2 n + c q a 2 n E (V q-1 (θ n )) + c q a q+1 n .
Then for any q ≥ 1 there exists A q ≥ 0 such that E[V q (θ n+1 )] ≤ a q n A q , ∀n ≥ 1. Proof of Lemma B.1. Initialization for q = 1: We prove the result by recursion on n. Let us first remark that V 0 (θ) ≤ e + V 1 (θ), then ( 14) becomes

E[V 1 (θ n+1 )] ≤ E (V 1 (θ n )) 1 -a n m + c 1 a 2 n + c 1 a 2 n E (e + V 1 (θ n )) + c 1 a 2 n ≤ E (V 1 (θ n )) 1 -a n m + 2c 1 a 2 n + c 1 (1 + e)a 2 n . (46) Now, consider some n 0 such that ∀n ≥ n 0 1 -a n m + c 1 a 2 n ≤ 1 - m 2 a n and a n a n+1 ≤ 1 + m 4 a n .
Let us first remark that for every fixed n 0 , we can choose A 1,n 0 large enough such that:

E(V 1 (θ n )) ≤ A 1,no a n , ∀n ≤ n 0 . Then E[V 1 (θ n+1 )] ≤ A 1,n 0 a n 1 -a n m 2 + c 1 (1 + e)a 2 n ≤ A 1,n 0 a n+1 a n a n+1 1 -a n m 2 + c 1 (1 + e) A 1,n 0 a n . Now if A 1,n 0 is large enough: E[V 1 (θ n+1 )] ≤ A 1,n 0 a n+1 a n a n+1 1 -a n m 4 ≤ A 1,n 0 a n+1 (1 + m 4 a n ) 1 -a n m 4 ≤ A 1,n 0 a n+1 ,
which concludes the recursion and prove the property for q = 1.

Recursion step: We assume that the property holds until the integer q -1 and we prove similarly by induction on n that the property holds for the integer q. Let us first remark that for every fixed n 0 , we can choose A q,n 0 large enough such that: E(V q (θ n )) ≤ A q,no a q n , ∀n ≤ n 0 . We remark that for n 0 large enough:

∀n ≥ n 0 a q n a -q n+1 ≤ 1 + m 4 a n and 1 -a n m + c 1 a 2 n ≤ 1 - m 2 a n .
Now, we apply Theorem 2.2 and conclude that

ω 2 n+1 E(θ n -θ α ) 2 (n) -4+a
. For the second term, we combine the bound on a 2 n (1/(N + 1) -n+1 ) obtained in (47) with Theorem 2.2 and deduce that:

a 2 n L 2 2 ( 1 n + 1 -n ) 2 E(θ n -θ α ) 4 ) n -2-2a .
Combining the two bounds, we deduce that:

E(R 2 n ) n -((4-a)∧(2+2a)) .
Proof of Theorem 1.1-ii). We use the key relationship introduced in [Pel00] derived from the linearization:

θ n+1 -θ n = -a n f (θ α )(θ n -θ α ) + a n R n + a n ∆M n+1 .
Dividing by a n and summing, we then obtain that:

n k=1 θ k+1 a k - θ k a k = -f (θ α ) n k=1 (θ k -θ α ) + n k=1 R k + n k=1 ∆M k+1 .
Finally, dividing by n and using an Abel transform, we verify that:

f (θ α )[ θn -θ α ] = 1 n θ 1 -θ α a 1 - θ n+1 -θ α a n+1 + n+1 k=2 1 a k - 1 a k-1 (θ k -θ α ) + n k=1 R k + 1 n n k=1 ∆M k+1 . (48)
The Jensen inequality applied to (48) yields:

| θn -θ α | 2p ≤ n -2p 5 2p-1   θ 1 -θ α a 1 2p + θ n+1 -θ α a n+1 2p + n+1 k=2 1 a k - 1 a k-1 (θ k -θ α ) 2p + n k=1 R k+1 2p + n k=1 ∆M k+1 2p  
We then compute the expectation of the right hand side terms and remark that:

(49) E θ 1 -θ α a 1 2p = O(1),
while Theorem 2.2 yields:

(50) E θ n+1 -θ α a n 2p = O(a -p n ) = o(n p ),
since a n = a 1 n -a with a ∈ (0, 1).

The generalized Holdër inequality yields (see e.g. Lemma 4.3 of [God15])

E n k=2 1 a k - 1 a k-1 (θ k -θ α ) 2p ≤ n k=2 1 a k - 1 a k-1 E|θ k -θ α | 2p 1/2p 2p . When a k = a 1 k -a , we can verify that 1 a k -1 a k-1 k -1+a
. Using again Theorem 2.2, we have

E|θ k -θ α | 2p 1/2p (a p n ) 1/2p = √ a n n -a/2 .
Therefore a comparison between series and integral implies that:

(51) E n k=2 1 a k - 1 a k-1 (θ k -θ α ) 2p n k=2 k -1+a/2 2p n ap .
Concerning the rest terms, we use that

|R k+1 | |θ k -θ α | 2 (see Proposition 2.4 iii))
, Theorem 2.2 and we deduce from the generalized Holder inequality that:

E n k=1 R k+1 2p ≤ n k=1 E|R k+1 | 2p 1/2p 2p n k=1 a 2p n 1/2p 2p n k=1 a n 2p n 2(1-a)p . ( 52
)
We remark that (∆M k+1 ) k≥1 is a bounded sequence of martingale increments. We shall apply the recursive argument stated in Lemma A.2 of [God15]: a constant K p exists such that:

(53) ∀n ≥ 1 E   n k=1 ∆M k+1 2p   ≤ K p n p .
We then gather Equations ( 49)-( 53) and obtain that: then a constant C α exists such that:

|η n | ≤ C α n -2-b+2a .
Proof. We recall from Section 2 that a large enough constant C α exists such that:

| n | ≤ C α na n f (θ α ) and | n -n+1 | ≤ C α n -2+a .
In the meantime, we can check that a large enough C α exists such that:

|δ n | ≤ C α θ α 1 -α n -b+a and |δ n -δ n+1 | ≤ C α n -1-b+a .
The triangle inequality yields:

|η n | ≤ | n+1 -n ||δ n+1 | + | n ||δ n -δ n+1 | ≤ C α n -2-b+2a .
( Z(n) 0 ) n≥1 = ( Zn ) n≥1 is a tight sequence.

To prove the tightness of ( Z(n) ) n≥1 , we follow Theorem 7.3 of [START_REF] Billingsley | Convergence of Probability Measures[END_REF] and we are led to study

P sup s∈[t,t+δ] | Z(n) s - Z(n) t | ≥
. This difference will be decomposed as:

Z(n) s - Z(n) t = B (n) s -B (n) t + M (n) s -M (n) t .
Step 2: Study of

B (n) s -B (n) t
-Tightness of (B (n) ) n≥1 . We consider > 0, δ > 0 and s ∈ [t, t + δ]. We use the triangle inequality and verify that: Ẽk .

B (n) s -B (
For the first term of the right hand side, we use the Tchebychev inequality and obtain that:

1 k Z k-1 ≥   -2 δ 2 .
We handle the rest term in the same way and control

N (n,t+δ)+1 N (n,t) Ẽk 2
. We apply Equation (37), the triangle inequality and the Minkowski inequality and deduce that:

N (n,t+δ)+1 N (n,t) Ẽk 2   N (n,t+δ)+1 N (n,t) | θk -θ α | k 3/2 + | θk -θ α | 2 √ k + |θ k -θ α | k 3/2-a + |θ k -θ α | 2 √ k + | ϑ k -ϑ α | k 3/2
| θk -θ α | 2 k 2 + | θk -θ α | 4 + |θ k -θ α | 2 k 2-2a + |θ k -θ α | 4 + | ϑ k -ϑ α | 2 k 2   .
We shall now compute the expectation of the previous terms and verify that

1 k Ek| θk -θ α | 2 k 2 + Ek 2 | θk -θ α | 4 k + k a E|θ k -θ α | 2 k 1-a + Ek 2a |θ k -θ α | 4 k 2a-1 + Ek| ϑ k -ϑ α | 2 k 2 .
Using now Theorem 2.2 (L p loss on the sequence (θ n ) n≥1 ), Theorem 2.2 (L p loss on the sequence ( θn ) n≥1 ) and Theorem 1.2 (L 2 loss on the sequence (ϑ n ) n≥1 ), we obtain that the bracket term is uniformly bounded in k, and therefore: We then apply the Tchebychev inequality and obtain that: Hence, for any > 0, for any η > 0, a small enough δ exists (of the order η 2 ) and a large enough n 0 exists such that:

∀n ≥ n 0 P   N (n,t+δ) N (n,t) Ẽk ≥   ≤ ηδ.
Theorem 7.3 of [START_REF] Billingsley | Convergence of Probability Measures[END_REF] implies that (B (n) ) n≥1 is a tight sequence of continuous processes.

Step 3: Study of

M (n) s -M (n) t
-Tightness of (M (n) ) n≥1 . Since s ∈ [t, t + δ], M (n) s safisfies:

M (n) s -M (n) t ≤ max( M (n) N (n,s) -M (n) t , M (n)
N (n,s)+1 -M

(n) t

).

Considering now the supremum over s ∈ [t, t + δ], we then deduce that:

sup s∈[t,t+δ] M (n) s -M (n) t ≥ ⊂ sup N (n,t)+1≤k≤N (n,t+δ)+1 M (n) Γ k -M (n) t ≥ .
We then deduce that for any p > 2:

M (n) s -M (n) t ≥ ≤ p p -1 p -p N (n,t+δ)+1 k=N (n,t) k -p/2 E ka k-1 ( 1 k -k )∆M k √ b 1 (1 -α) k+1 k ∆N k p
First, the martingale increments (∆M n ) n≥1 are uniformly bounded (see Equation ( 8)). Second, from our assumption H f , the random variable X possesses a moment of order p > 2 and for this value of p, we observe that:

P sup s∈[t,t+δ] M (n) s -M (n) t ≥ -p N (n,t+δ)+1 k=N (n,t) k -p/2 .

E|

  θn -θ α | 2p n -2p [O(1) + O(n p ) + O(n ap ) + O(n 2(1-a)p ] = O(n -p ), for a ∈ [1/2, 1), which concludes the proof. Appendix C. Technical results for the super-quantile algorithm ( ϑ n ) n≥1 This paragraph gives the missing details of the proof of Proposition 3.3. Lemma C.1. Assume that the step-size sequences satisfy a n = a 1 n -a and b n = b 1 (n + 1) -b with 1/2 < a < b ≤ 1.

N

  (n,t+δ)+1 N (n,t) 1 k Z k-1 ≥   ≤ -2 E   N (n,t+δ)+1 N (n,t) Since sup k≥1 E Z k 2 < +∞ and N (n,t+δ)+1 N (n,t)1 k ≤ 2δ for large n, we then obtain that:

.

  Now we apply the Cauchy Schwarz inequality in order to useN (n,t+δ)+1 N (n,t) 1 k in each term: N (n,t+δ)+1 N (n,t)

Γ

  N (n,t+δ)+1 -M (n) t p ,where we first apply the Markov inequality and second the Doob maximal inequality. Now, the Minkowski inequality yields:P sup s∈[t,t+δ]

S. G. acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (Investissements d'Avenir, grant ANR-17-EURE-0010). S. G. acknowledges funding from the French National Research Agency (ANR) for the grant MaSDOL -19-CE23-0017-01. M. C. and S. G. would like to thank Bernard Bercu for fruitful discussions about their work.

Appendix A. From Lyapunov contraction to non-asymptotic upper bound

The next lemmas provide non-asymptotic upper-bounds from a Lyapunov-type contraction.

Lemma A.1. Let b n = b 1 n -b with 1 2 < b < 1, A, C > 0, r 1 > 3b 2 and r 2 > 2b. Consider a positive sequence u n such that:

then, for any p ∈]b, (r 1 -b/2) ∧ (r 2 -b) ∧ 2b ∧ 1[, a large enough Γ p exists such that:

(42)

Proof. We prove the result with a recursive argument. The initialization is obtained for a large enough Γ. Assume that Equation (42) holds true for an integer n, then we have

where:

n , and:

The aim is now to prove that for a good choice of p, e n+1 + ẽn+1 is negative. For e n+1 an easy computation leads to:

For ẽn+1 , we combine the two first terms to obtain:

This difference is therefore negative as soon as n is large enough. Finally:

, then a large enough Γ exists such that e n+1 + ẽn+1 ≤ 0. It concludes the proof.

We now consider the limit case where r 1 = 3/2. Lemma A.2. Let γ > 1/2, A 1 , A 2 , C > 0, r 1 > 3/2 and r 2 > 2 and consider a positive sequence u n such that

Let us define:

Remark 2. Using the value of V , we have

Therefore the condition ρ-2γ+A 1 /(2 √ V ) < 0 is automatically satisfied as soon as γ > ρ-1/2.

Proof. We proceed by induction on n, in order to prove that u n ≤ V n + Γn -ρ , for the largest possible value of ρ. The initialization is obvious at the price of a large enough Γ. We consider the recursion property assuming that the property holds at iteration n. Then:

We observe that for n large enough, we have:

Hence, we deduce that:

We choose √ V as the positive root of (1 -2γ)X 2 + A 1 X + C and obtain that:

(43)

for which the leading term of n vanishes. Then, we observe that

Hence, n is upper bounded by:

Since r 1 > 3/2 and r 2 > 2, we then observe that as soon as (ρ, γ) satisfies

then a large enough Γ exists such that n ≤ 0.

Appendix B. Technical results for the quantile estimation

Proof of Lemma 2.3. A straigthforward computation from the Equation (9) yields:

(44)

Proof of i): Using (44) we deduce that

and we derive a lower bound for the bracket term q Φ (θ) 2 Φ(θ) + Φ (θ) 2 on R. Using the definition (6), we know that Φ (θ α ) = f (θ α ) > 0 and in a neighborhood of θ α :

since Φ (θ α ) = 0. We deduce that an ε > 0 exists such that:

Using Equation ( 14), and the recursion assumption for q -1 we obtain that for n ≥ n 0 :

Therefore, up to a modification of the constant A q,n 0 :

We conclude that the property holds for all n ≥ n 0 and thus for any integer q:

B.2. Technical results for the averaged quantile estimation ( θn ) n≥1 . This paragraph gathers technical results useful for the study of ( θn ) n≥1 .

Proof of Lemma 2.5. A straigthforward computation yields:

We then conclude that a large enough constant C exists such that:

(47) a 2 n n+1 -

We recall that from Proposition 2.4:

Then, the Cauchy-Schwarz inequality yields E|θ

α(1 -α) + Cn -a/2 , from which we deduce the lemma.

The second lemma handles the rests terms given in (20).

Proof of Lemma 2.6. Using the Jensen inequality as well as Proposition 2.4 iii), we deduce that:

For the first term, a direct computation shows that:

We now prove the two lemma contained in the proof of Proposition 3.3.

Proof of Lemma 3.4. Recall that U n defined in (29) is the sum of four terms so that:

We consider each term of the previous sum separately. We remind that O

(1) n = θ n -θ α . Then, Lemma C.1 and Theorem 2.2 yield:

Concerning the second term, remark that O

(2)

n , so that (21) entails:

For the rest term R n , we use that: R n ≤ L 2 |θ n -θ α | 2 and Theorem 2.2, so that:

Finally, for the last term, we use that

Then, Theorem 2.2 leads to:

. Gathering all the previous bounds and using 1/2 < a < b ≤ 1, we get:

Proof of Lemma 3.5. We study the variance terms brought by ∆V n :

and study each term separately. First term. From Lemma 2.5, E({∆M n+1 } 2 ) ≤ α(1 -α) + Cn -a/2 . Combining this with the upper bound of n δ n = n -2+2a in Lemma C.1, we observe that:

Second term. We use (25) and Theorem 2.2 to deduce that:

Third term. We verify with simple algebra that:

We apply the Cauchy-Schwarz inequality and obtain that:

We deduce that a large enough constant exists such that:

In particular, we verify that:

We obtain with simple algebra using Lemma C.1 that:

).

To conclude we gather the three upperbounds and deduce that:

Appendix D. Technical results for the central limit theorem

Proof of Proposition 4.2. Proof of i): The proof relies on Theorem 7.3 of [START_REF] Billingsley | Convergence of Probability Measures[END_REF] and is divided into three steps.

Step 1: Tightness of ( Z(n) 0 ) n≥1 . We shall remark that

We observe that Tn = √ n Z

(2)

n and (21) implies:

n } 2 < +∞. In the meantime, we deal with the initialization of V n by applying Theorem 1.2. We then deduce that: sup n≥1 EV 2 n < +∞, so that sup n≥1 E Zn 2 < +∞. We then conclude that Hence, we obtain that:

We conclude that for this value of p > 2, for any > 0 and any η > 0, a n 0 exists such that:

which in turn implies that:

Again, Theorem 7.3 of [START_REF] Billingsley | Convergence of Probability Measures[END_REF] implies that (M (n) ) n≥1 is a tight sequence of continuous stochastic processes. It concludes the proof of the i).

Proof of ii):

We start from the definition of ∆M n+1 given in Proposition 4.1:

We now study each term separately.

• Lemma 2.5 implies that:

• Proposition 3.1 yields:

• Finally, we deduce from Lemma 3.5 iii) that S 1,2 n = a n (n + 1)(

Using our previous results on (θ n ) n≥1 and ( θn ) n≥1 , the conclusion holds. Proof of iii): We consider a twice differentiable test function ϕ. Using a second order Taylor expansion and Proposition 3.1, we obtain that:

where G(ϕ) is defined in Equation (39). Finally, combining (37), Theorem1.1, Theorem 1.2 and the fact that the martingale increments are bounded, we deduce that E((n + 1)|Q n |) -→ n→∞ 0.


Stéphane Pouyllau

Mélanie Bunel

Laurent Capelli

Jean-Luc Minel

"We": a Proposal for the TRIPLE platform

Keywords: TRIPLE project, discovery platform, experts, SSH tools Important

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The early 2000s saw the production of a very large number of studies on the analysis of researchers access practices to bibliographic information sources and documents (publications, serial data, etc.). In their study conducted in 2008 [START_REF] Ihadjadene | Quelles analyses de l'usage des moteurs de recherche[END_REF], the authors identified more than a hundred such studies. In their conclusion, the authors insisted on a point that seems still relevant today: "A search engine is no longer the "place" where different cognitive structures are matched in interactions, but is seen as a more global system in which multiple variables come into play: the cognitive space of the actors, psychological, social and organizational contextual characteristics, and changing information needs. It is important to apprehend the user in an information-seeking situation in a much more global way than they are studied in cognitive models and, a fortiori, in the approach which still often underlies current studies on the use of search engines". 1In the last ten years, after the arrival of Google Scholar, the number of current platforms has risen sharply. The study conducted by Know-Center, the TRIPLE WP7 leader [START_REF] Breitfuss | Deliverable D7.1, Report on Stakeholder and Opportunity Analysis TRIPLE Project[END_REF], lists 47 platforms, with the arrival of private (ResearchGate, Academia) or nongovernmental (Semantic Scholar2 , Dimensions3 ) stakeholders. Some of these projects are led by TRIPLE's partners (for example with Viper4 developed by Open Knowledge Maps for OpenAIRE or Veille.science developed by Huma-Num with the search engine ISIDORE [START_REF] Pouyllau | Veille.science : une veille et intelligence scientifique en SHS à destination des décideurs[END_REF] 5 ). This study [START_REF] Breitfuss | Deliverable D7.1, Report on Stakeholder and Opportunity Analysis TRIPLE Project[END_REF] corroborates the results published in 2016 by [START_REF] Lopez-Cozar | Google Scholar as a data source for research assessment[END_REF], namely that Google Scholar is the search engine used by 89% of users. Similarly, in the study carried out by a partner (Abertay University) of the TRIPLE project for deliverable D3.1 [START_REF] Forbes | Deliverable D3.1, Report on user needs[END_REF], Google Scholar is the search engine used by the majority of users interviewed. At the same time, usage studies have practically disappeared from the research work.

Without denying the importance of needs studies or usage studies [START_REF] Forbes | Deliverable D3.1, Report on user needs[END_REF], our proposals are rather based on feedback and targeted analysis of the ISIDORE platform's6 uses as well as on the study of different platforms, such as Semantic Scholar or Dimensions. It should be noted that all these platforms claim to be based on technologies derived from Artificial Intelligence and more specifically from Deep Learning. Our proposal thus attempt to position a socio-technical device [START_REF] Akrich | Les formes de la médiation technique[END_REF][START_REF] Latour | La Clé de Berlin. Petites leçons en sociologie des sciences[END_REF] different from the existing platforms by specifying new functionalities in a complementary and non-competitive way.

General principle of the We platform ("We" for "Who are the experts") is to boost the discovery of scientific issues in the fields of SSH by exploring connections between scholars (the experts), their scientific tracks (or evolutionary tracks of their work) and their productions (projects, publications, etc.). These tracks are the result of computations relying on Machine Learning techniques. They requires the commitment of the project partners to maintain efficiently and regularly updated semantic annotations. The We platform is part of development of "deep FAIR platforms" [START_REF] Parland-Von Essen | 2nd Report on FAIR requirements for persistence and interoperability (D2.4)[END_REF].

The We platform is positioned as the main access platform to the TRIPLE ecosystem. Accessible via a website, it provides access to information on the TRIPLE project, experts and information search tools in Social Sciences and Humanities (SSH) and value-added services developed by TRIPLE partners. Functioning as a "hub", it is in fact the main entry point for TRIPLE users. This research report presents the lines of work of the Software Design and Information Management Team (GLIST) from Huma-Num 7 in terms of definition of functionalities of the We platform. The proposals are largely inspired from the work carried out for the design and implementation of the ISIDORE platform in France opened in 2010. In relation to the functional architecture [START_REF] Bunel | Thinking about the Architecture of the Discovery Platform of the TRIPLE Project[END_REF], this report deals with the perimeter surrounded by the green rectangle in the figure 1. 2 The Big Picture

Target Audience

In line with the TRIPLE proposal [START_REF]Transforming Research through Innovative Practices for Linked interdisciplinary Exploration TRIPLE. The European discovery platform dedicated to SSH resources[END_REF], the We platform is intended for :

• Scholars and PhD students from the SSH;

• Journalists (in order to diversify their sources, experts, etc.); 7 Huma-Num is a French Very Large Research Infrastructure with international reach devoted to Social Sciences and Humanities. See https://www.huma-num.fr

• Audiences seeking access to reliable information based on the notion of scientific community (include digital humanists who need advanced tools). For detailed audience, see the TRIPLE Deliverable: D8.3 Communication Strategy. 8It can be accessible via a sub domain URL of gotriple.eu (such as for example we.gotriple.eu).

Definition

The We platform is positioned as a socio-technical device [START_REF] Akrich | Les formes de la médiation technique[END_REF][START_REF] Akrich | The De-Scription of Technical Objects[END_REF][START_REF] Latour | La Clé de Berlin. Petites leçons en sociologie des sciences[END_REF] dedicated to the discovery of complementary scientific issues compared to generalist academic search engines (OpenAIRE, Google Scholar, Semantic Scholar, Dimension) but also more sectorial or disciplinary (such as ISIDORE for the SSH) or even compared to the discovery platforms provided to university libraries [START_REF] Bourdenet | L'espace documentaire en restructuration : l'évolution des services des bibliothèques universitaires[END_REF] by scientific publishers.

It exploits, via the TRIPLE API/REST, the enrichments and semantic links produced by applying Machine Learning methods in order to editorialize the contents of the TRIPLE database. In order to create semantic relationships between informational objects of different natures and coming from several European data aggregators, potential of data linking based on the principles of Linked Open Data (LOD) [START_REF] Berners-Lee | Linked Data, Design Issues[END_REF] are also exploited. As such, it does not take the form of a Web portal for documentary research, but offers discovery and navigation between scientific concepts ("Topics") and experts (see 4) in the sense of a set of research topics studied by a scholar. It should be noted that the linking is carried out upstream of TRIPLE's API/REST on the basis of a fine indexing of the metadata, of the full text when it is freely available.

In a nutshell, the We platform provides access to research data (publications, documents, skills, current or past projects, etc.) through an editorialization of the notion of experts and groups of them in SSH fields derived from TRIPLE's semantic processing 9 . In this way, We platform fully exploits the enrichment processing chain of TRIPLE [START_REF] Bunel | Thinking about the Architecture of the Discovery Platform of the TRIPLE Project[END_REF].

Architecture

The We platform brings together on a single website all the services and tools giving access to enriched information as well as information relating to the life of the project. It contains (fig. 2):

• The information flow on TRIPLE project news (for example via RSS feeds from a TRIPLE Web blog);

• A gateway, relying on SSHOC catalogue10 , giving access to descriptions of valueadded services (via a database describing these services). To be included in this registry, a service must be generic enough and freely reusable for other platforms11 ;

• Several web pages 12 enabling the discovery of scientific issues in the SSH fields. These pages contain all or part of TRIPLE's visualization, annotation, recommendation services, either in the form of an inclusion in We's Web UI, or as links pointing to a third party UI. We detail below the functionalities offered by these web pages (fig. 3).

• The ability to search, find and browse the work, activities of scientific experts;

• The ability to find projects and documents related to experts in one or more fields of research (see "Topics" below);

• The possibility to identify rising scientific emerging issues, innovative or early stage work, by using computed enrichment and networks of experts. The We platform will integrate different services such as visualization, recommendations, and annotations developed by the WP5 led by OKMaps/Nuromedia, and Net7, partners.

Theses services will undoubtedly have to be accessible in the form of API results in order to be used in the Web interface designed by the WP4. These integrations will depend both of WP3 and WP5 13 . There are two modes of use of the WE platform, an unconnected mode (fig. 4) and a connected mode (fig. 5), with a specific web page for each mode. 3 Topics Box "TopicsBox" (fig. 6) are boxes in which a user specifies a scientific query for which he is looking for experts. The "TopicsBox" are the heart of access to TRIPLE's enriched information and the mandatory entry point to the rest of the discovery system. A scientific query combined concepts selected and edited by the partners of the TRIPLE project. Borrowed from scientific news or from the thesaurus 14 Library of Congress Subject Headings (LCSH) of the Library of Congress which are potentially related to several other concepts from repositories expressed in the LOD and aligned on Wikidata. The editorialization of the TopicsBox is under the responsibility of the WP3 with the support of the GLIST team.

Building TopicsBox in the Browser

TopicsBox are built by aggregating concepts searched in the TRIPLE database via the API/REST. An registered user can manage his TopicsBox in his library. We want to point out that TopicsBox combined topics (concepts) (fig. 7) provided by the TRIPLE API, consequently an user can not express a query with his own keywords.

Experts

Looking for an Expert

The notion of expert has given rise to an abundance of literature [START_REF] Bessy | Experts et faussaires. Pour une sociologie de la perception, Editions Pétra, coll[END_REF] as well as finding experts in academics [START_REF] Angelova | An expertise Recomender System Based on Data from Institutional Repository (DiVA)[END_REF][START_REF] Cifariello | Wiser: A semantic approach for expert finding in academia based on entity linking[END_REF]. A survey on the state-of-the-art methods is presented in [START_REF] Lin | A Survey on Expert Finding Techniques[END_REF]. There are several reasons for using this type of system such as: recruiting reviewers, looking for scientific partners for research proposals, finding a PhD supervisor, and so on.

As described in [START_REF]Transforming Research through Innovative Practices for Linked interdisciplinary Exploration TRIPLE. The European discovery platform dedicated to SSH resources[END_REF][START_REF] Forbes | Deliverable D3.1, Report on user needs[END_REF] looking for experts, relying mainly on university open archives, is a crucial task whose goals cover a wide range of needs in view of the target audience: scholars, students, journalists, and so on. As pointed out by [START_REF] Angelova | An expertise Recomender System Based on Data from Institutional Repository (DiVA)[END_REF] "The use of university repositories is not so common, in spite of their wealth of bibliographic metadata of both local scientific records and student theses records. (...) One of the reasons for not exploiting institutional repositories on a greater scale could well be that they are often created, administered and run by librarians who generally are not research oriented and especially not towards computer science. Another reason might be that the data in the repositories often is entered by researchers and students themselves and therefore, in many cases, are not quality controlled."

In the We platform, an expert is a highly active scientist who published scientific papers, produced scientific data sets or contributed to scientific projects. More specifically, a TRIPLE expert is defined by a certain volume of enrichments (disciplines, concepts) made from the semantic processing of the metadata of his publications, projects and description elements of his profile (description in his CV, etc.) and potentially from the processing of his scientific papers (depending on its availability).

A TRIPLE expert is characterized first of all by his/hers disciplines of expertise based on disciplinary categorization and annotation of his publications and/or projects. Then, his/hers links with other experts from the same disciplines and having one or more themes in common. These links may be upstream or downstream: "has been influenced by ...", "has influenced ..." or "is cited by ...". Finally, an expert belongs to a community (see below the notion of group) having a sustained and/or regular co-publishing activity.

An expert can thus be characterized by a set of indicators that can be provided or computed by TRIPLE :

• An indicator of activity based on publication activity, membership of journal editorial boards ;

• An indicator of notoriety based on citations of the researcher's work.

• An indicator of participation in communities based on his/her participation in research projects.

The objective of this report is not to detail algorithms used to calculate a scholar's degree of expertise, neither how to rank them, but the prototype DIVA proposed by [START_REF] Angelova | An expertise Recomender System Based on Data from Institutional Repository (DiVA)[END_REF] could be a realistic source of inspiration. It should be noted that the computing of these indicators requires access to data not currently provided by aggregators (Isidore, OpenAire, Mosa, Narcis, etc.). Therefore, if these functionalities were retained, it would be necessary to engage discussions with these aggregators.

Displaying Experts

In the We platform, looking for experts is always conditioned by the expression in the query of topics and/or disciplines. Figure 8 shows an example of a sketch of screen dedicated to perform this kind of query and figures 9 and 10 propose some sketches showing results of this query. More precisely, the screen in figure 9 displays co-authors and publications of an expert, as well as projects in which the expert is or was involved. It should be noted that the links to publications are organized by aggregators (Isidore, OpenAIRE, Narcis, etc.) which allows the user to have a first appreciation of the expert's work. The screen in figure 10 displays projects in which the expert is or was involved and a list of scholars who influenced him. It should be noted that these data about experts could be enriched by exploiting data not provided by TRIPLE. For example, these complementary data can be taken from different services mentioned by [START_REF] Fenner | Introducing the PID Graph[END_REF] such as Research Graph Foundation15 , Scholix16 [START_REF] Burton | The Scholix Framework for Interoperability in Data-Literature Information Exchange[END_REF], a framework for Scholarly Link Exchange coordinated by a Research Data Alliance (RDA) Working Group, and the OpenAIRE Research Graph [START_REF] Manghi | The OpenAIRE Research Graph -Opportunities and challenges for science[END_REF]. The FREYA project17 mentioned in the TRIPLE WP6 led by Net7 partner [START_REF]Transforming Research through Innovative Practices for Linked interdisciplinary Exploration TRIPLE. The European discovery platform dedicated to SSH resources[END_REF] should become an important resource which should be used by TRIPLE.

Proposal of Innovative Concepts and Applications

Emerging Themes

As stated above, principles of the We platform is to boost the discovery of scientific issues in the fields of SSH through the links between scholars who are experts in topics, belong to well-known research communities (the groups) and their scientific tracks (or evolutionary track of their work) and their scientific productions (projects, publications, data sets, etc.). The notion of scientific tracks potentially leads to the notion of "emerging themes" or "weak signals" bringing together several experts who do not necessarily have a link based on the co-signature of publications. Belong to an "emerging theme" induces connections that are based solely on concepts, disciplines and any common enrichment between experts who are not co-authors. This notion is particularly interesting in the linguistic dimension of TRIPLE enrichment (9 languages). Discovering experts, who have never published together, from a pool of documents, projects or profiles, in 9 languages is one of the innovative and unprecedented features in the specialized field of SSH.

Looking for emerging themes requires taking into account the evolution over time of the experts's research themes as well as the identification of groups to which they belong. There are several ways to identify experts on emerging themes. One approach would be to rely on Deep Learning techniques by exploiting open source machine learning framework such as Pytorch18 or Keras 19 . More specifically, the classification of authors' titles and/or abstracts is an approach to be explored. In the same way as for the TopicsBoxes (see above), TRIPLE could offer the integration of a Jupyter Notebook for experienced users (detailed in Section 5.3).

Groups of Experts

Expert groups bring together experts from different disciplines who are not co-authors but work on related themes. An example of the representation of these groups and for an emerging theme is shown in Figure 11. We think that some propositions made by TRIPLE partner Open Knowledge Maps (OKMAPS) could be used and/or adapted to carry out this kind of searching. This suggestion needs further investigations which may be discussed later.

Building Embedded TopicsBox

With the idea of targeting digital humanists among the potential users of the platform, we propose to create a specific mode in order to editing "TopicsBox" in a programming environment in the form of Python libraries (or other) packaging the API/REST notebooks under the Jupyter environment with Python packages. Indeed, these environments are increasingly being used in the field of data science [START_REF] Herta | Deep Teaching: Materials for Teaching Machine and Deep Learning[END_REF] and in the construction of data analysis applications, particularly in higher education [START_REF] Pouyllau | L'exploitation d'ISIDORE et de son SPARQL endpoint avec Jupyter[END_REF]. This functionality will open up important possibilities for researchers mastering these programming environments.

For example, figures 12 and 13 show two Python scripts and their results that can be performed from the Triple SPARQL EndPoint via a Jupyter Notebook. The first example calculates the disciplines in which an author publishes and represents it with a pie chart. The second one computes the number of co-publications from an author 20 . Some plateforme use Python packages in Jupyter Notebooks, for example, DataCite, with GraphQL and PID Graph [START_REF] Fenner | Using Jupyter Notebooks with GraphQL and the PID Graph[END_REF].

Our hypothesis is that a large number of scholars and PhD students master these techniques and that TRIPLE provides them a massive source of data on which it is possible to discover and process new research objects [START_REF] Jacob | Pour une herméneutique numérique en sciences historiques[END_REF][START_REF] Pouyllau | L'exploitation d'ISIDORE et de son SPARQL endpoint avec Jupyter[END_REF][START_REF]Digital Technology and the Practices of Humanities Research[END_REF].

Conclusion

We propose a socio-technical device, called We platform, to highlight and exploit the work of data collection, aggregation and enrichment carried out for many years by national search engines in the SSH disciplines or pan-continental portals who propose institutional sets.

This platform editorializes semantic enrichments carried out by the TRIPLE project by proposing the discovery of scientific issues in the fields of SSH by taking advantage of links between scholars (the experts), their known research communities (the groups) and their scientific tracks (or evolutionary track of their work) and their productions (projects, publications, etc.). This platform takes advantage of various services developed by TRIPLE project partners by including them either in a catalogue of services or directly in the HMI, i.e. in the User Interface Design (UI) to serve the User Experience Design (UX).

Figure 1 :

 1 Figure 1: Functional architecture of TRIPLE and the We platform scope (diagram by M. Bunel. Source: Adapted from Deliverable 2.1: Data Acquisition Plan, 2020, TRIPLE project)

Figure 2 :

 2 Figure 2: Simplified architecture of the We platform and its services (diagram by S. Pouyllau)

Figure 3 :

 3 Figure 3: Detail of the functional pages of the We platform (diagram by S. Pouyllau)

Figure 4 :Figure 5 :

 45 Figure 4: General picture for unconnected users (drawing by S. Pouyllau)

Figure 6 :

 6 Figure 6: General picture of the concept of "TopicsBox" (drawing by S. Pouyllau)

Figure 7 :

 7 Figure 7: Editor of "TopicsBox" (drawing by S. Pouyllau)

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Looking for experts (sketch from S. Pouyllau)

Figure 11 :

 11 Figure 11: Expert groups and emerging themes (sketch from S. Pouyllau)

Figure 12 :Figure 13 :

 1213 Figure 12: Computing an author's fields from the TRIPLE SPARQL EndPoint in a Jupyter notebook (screenshot from S. Pouyllau)

Translated from French by the authors of this report.

https://www.semanticscholar.org

https://www.dimensions.ai

https://openknowledgemaps.org/viper/

"Veille.science" is a proof of concept developed in 2016 around ISIDORE, see https://doi.org/10. 5281/zenodo.3965604. Some elements are reused in the this proposal.

https://isidore.science

https://zenodo.org/communities/operaseu/

Specific plateforms, like Association for Computing Machinery (https://dl.acm.org/people) propose interesting examples.

[START_REF] Bunel | An ontology for the TRIPLE Data Model[END_REF] https://sshopencloud.eu/ssh-open-

marketplace[START_REF] Bunel | Thinking about the Architecture of the Discovery Platform of the TRIPLE Project[END_REF] It will depend on the policy of the TRIPLE project partners and on the positioning of these tools after the funded period

of the project.[START_REF] Bessy | Experts et faussaires. Pour une sociologie de la perception, Editions Pétra, coll[END_REF] Named w1, w2, ... in figure3

It should be noted that at the time of writing this document, it is not possible to specify the modalities of integration in the Human-Machine Interface (HMI) We platform

The frequency of concepts updating will be decisive for the We plateform.

https://researchgraph.org

http://www.scholix.org

FREYA is a 3-year project funded by the European Commission under the Horizon 2020 programme. The project aims to build the infrastructure for persistent identifiers as a core component of open science, in the EU and globally. FREYA will improve discovery, navigation, retrieval, and access of research resources https://www.project-freya.eu/en

https://pytorch.org

https://keras.io

Full examples are available and machine readable with Binder for Jupyter notebooks at https:// gitlab.huma-num.fr/spouyllau/isidore-jupyter/

We : A Proposal Glossary GLIST Software Design and Information Management Team. 3 HMI Human-Machine Interface. 6, 14 LCSH Library of Congress Subject Headings. 7 LOD Linked Open Data. 4, 7 SSH Social Sciences and Humanities. 2, 11, 14 UI User Interface Design. 14 UX User Experience Design. 14


Stress, Drug Resistance, Adhesion:

the dark side of the wall.

Hélène Martin-Yken e-mail: helene.martin@insa-toulouse.fr e-mail: helene.martin@insa-toulouse.fr

Yeasts

-Eukaryotic single-cell microorganisms, members of the Fungus kingdom.

-Used by mankind for milenia to ferment food, notably Saccharomyces cerevisiae, S. bayanus, Kluveyromyces lactis...

-Life threatening pathogenic yeast species : Candida albicans and Cryptococcus neoformans.

  images of yeasts cells (source: Chemistryland.com).

  to unravel the morphologic effect on yeast cells of Heat Shock, Osmotic Shock, exposure to toxins and drugs.New projects include effect of mycotoxins on yeast cells, and molecular determinants of cell adhesion,-at the molecular scale with AFM and Optical Tweezers, -at the population scale with the Sheer Flow Chamber.