domain: stat info

text:
Maxime Haddouche

Paul Viallard
email: paul.viallard@

Umut Şimşekli

Benjamin Guedj
email: benjamin.guedj@

Maxime Haddouxhe

Umut Simsekli
email: umut.simsekli@

© M Haddouche

A PAC-Bayesian Link Between Generalisation and Flat Minima

Keywords: Generalisation Bounds, PAC-Bayes, Flat Minima, Poincaré, Log-Sobolev Inequalities

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

Introduction

Understanding generalisation in modern machine learning problems has been a major challenge in learning theory. The goal here is to upper-bound the so-called generalisation error that is gap between the population and empirical risks, R D (h) -RSm (h), where h ∈ R d is the parameters of a predictor, R D := E z∼µ [ℓ(h, z)] is the population risk, D is an unknown data distribution, ℓ is a loss function, RSm := 1 m m i=1 ℓ(h, z i ), and finally S m := {z 1 , • • • , z m } is a dataset with each z i is independent and identically distributed (i.i.d.) with D.

Dating back to [START_REF] Hochreiter | Flat minima[END_REF], it has been hypothesised that the notion of 'flatness' (or sometimes equivalently referred to as 'sharpness') has tight links with the generalisation error: among the minima (belonging to RSm ) that is found by the learning algorithm, the 'flatter' the minimum is, the lower is the generalisation error. While the initial flatness notion was (vaguely) defined through low Kolmogorov complexity, there is no single formal definition of 'flatness'. Hence, several flatness notions have been considered, which typically are based on the second-order derivatives of the empirical risk around the local minimum found by the algorithm, such as trace (∇ 2 RSm (h)), see e.g., [START_REF] Jastrzebski | Three factors influencing minima in sgd[END_REF]; [START_REF] Wen | Sharpness minimization algorithms do not only minimize sharpness to achieve better generalization[END_REF].

While there have been several attempts to link some form of flatness to generalisation in a mathematically rigorous way [START_REF] Behnam Neyshabur | Exploring Generalization in Deep Learning[END_REF][START_REF] Petzka | Relative flatness and generalization[END_REF][START_REF] Yue | Sharpnessaware minimization revisited: Weighted sharpness as a regularization term[END_REF][START_REF] Andriushchenko | A modern look at the relationship between sharpness and generalization[END_REF], mainly in the framework of 'sharpness aware minimisation' [START_REF] Foret | Sharpness-aware minimization for efficiently improving generalization[END_REF], it has been recently shown that flat minima do not always imply good generalisation. In fact, there exist scenarios such that the flattest minima achieve the worst generalisation performance compared to non-flat ones [START_REF] Wen | Sharpness minimization algorithms do not only minimize sharpness to achieve better generalization[END_REF].

In this study, we aim at developing novel links between flatness and the generalisation error from a PAC-Bayesian perspective (see e.g., [START_REF] Guedj | A primer on PAC-Bayesian learning[END_REF][START_REF] Hellström | Generalization bounds: Perspectives from information theory and PAC-Bayes[END_REF][START_REF] Alquier | User-friendly Introduction to PAC-Bayes Bounds[END_REF]. Denoting by Q, the probability distribution of the algorithm output h (or the output of a learning algorithm), we identify sufficient conditions on Q such that flatness always implies good generalisation. More precisely, we make the following contributions:

• We show that, when Q satisfies the Poincaré inequality and a technical condition that we identify, we can obtain a 'fast-rate' generalisation bound that diminishes with rate 1 /m (rather than 1 / √ m) and mainly contains two terms:

(i) The flatness term:

E h∼Q 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 .
This term is directly linked to the Hessian of the loss ℓ, due to the connection between the Fisher information and the Hessian of the loss [START_REF] Peter | Mathematical statistics: basic ideas and selected topics, volumes I-II package[END_REF]. For instance, under certain conditions, it can be shown that trace (∇ 2 RSm (h)) = 2 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 (Wen et al., 2023, Lemma 4.1).

(ii) The classical PAC-Bayesian complexity term KL(Q, P), where KL denotes the Kullback-Leibler divergence and P is data-independent 'prior' distribution.

• We then further analyse the term KL(Q, P). We show that, when Q is a Gibbs distribution, i.e., Q(h) ∝ exp(-γ RSm (h))P(h) for some γ > 0 and P satisfies a log-Sobolev inequality, the generalisation error can be controlled solely by the term:

γ 2 c LS (P) E h∼Q [∥∇ h RSm (h)∥ 2 ],
where c LS (P) denotes the log-Sobolev constant of the prior P.

• We finally go beyond the KL divergence to link flat minima to deterministic predictors (i.e., when Q is a Dirac distribution) through a novel Wasserstein-based generalisation bound for gradient Lipschitz loss functions.

We provide a numerical assessment of the technical condition underlying our main result, suggesting that it is suitable in the case of neural networks on classification tasks, confirming the relevance of our bounds to better understand the generalisation ability of neural networks. Our results shed further light on the impact of the flatness of the minima over the generalisation error: when the learning algorithm ensures a sufficiently regular distribution over the parameters, the generalisation error can be directly controlled by the flatness of the region found by the algorithm.

Preliminaries

Framework. We consider a predictor set H ⊆ R d equipped with a norm ∥.∥, a data space Z and the space of distributions over H ∈ M(H). We also consider a loss function ℓ : H × Z → R. We assume that we have access to a i.i.d.

dataset S = (z i ) i≥1 ∈ Z N with associated distribution D. For each m ≥ 1, we define S m := {z 1 , • • • , z m }.
In PAC-Bayes learning, we construct a data-driven posterior distribution Q ∈ M(H) with respect to a prior distribution P . To assess the generalisation ability of a predictor h ∈ H, we define the population risk to be R D (h) := E z∼µ [ℓ(h, z)] and for each m, its empirical counterpart RSm (h

) := 1 m m i=1 ℓ(h, z i ).
As PAC-Bayes focuses on elements of M(H), we also define the expected risk and empirical risks for

Q ∈ M(H) as R D (Q) := E h∼Q [R D (h)] and RSm (Q) := E h∼Q [ RSm (h)]
. PAC-Bayes bounds usually aim at controlling the expected generalisation error (or gap) for each dataset size m, i.e., ∆ Sm (Q

) := R D (Q) -RSm (Q).
Background on Poincaré and log-Sobolev inequalities. In this work, we exploit Poincaré and log-Sobolev inequalities in the PAC-Bayes framework. We first recall the definition of Poincaré and log-Sobolev inequalities. To do so, for a fixed distribution Q, we define the Sobolev space of order 1 on R d as follows:

H 1 (Q) := f ∈ L 2 (Q) ∩ D 1 (R d ) | ∥∇f ∥ ∈ L 2 (Q) ,
where D 1 (R d ) denotes the set of derivable functions f : R d → R.

Definition 1 (Poincaré and Logarithmic Sobolev inequalites)

A measure Q satisfies a Poincaré inequality with constant c P (Q) if for all function f ∈ H 1 (Q) we have Var Q (f ) ≤ c P (Q) E h∼Q ∥∇f (h)∥ 2 , where Var Q (f ) = E h∼Q [f (h) -E h∼Q [f (h)]] 2 is the variance of f w.r.t. Q. We then say that Q is Poincaré with constant c P (Q), or that Q is Poinc(c P ). Also, Q satisfies a log-Sobolev inequality with constant c LS (Q) if for all function f ∈ H 1 (Q) we have E h∼Q f 2 (h) log f 2 (h) E h∼Q [f 2 (h)] ≤ c LS (Q) E h∼Q ∥∇f (h)∥ 2 ,
where the term on the left hand side is the entropy of f 2 , denoted as

Ent Q (f 2 ). We then say that Q is log-Sobolev with constant c LS (Q), or that Q is L-Sob(c LS ).
The class of Gaussian distributions is an important particular case of distributions satisfying both Poincaré and log-Sobolev inequalities, this is the subject of Proposition 2.

Proposition 2 For a given pair (µ, Σ) of mean and covariance matrix in R d , define Q = N (µ, Σ).

Then we have, for any f ∈ H 1 (Q):

Ent Q (f 2 ) ≤ 2E Q [⟨Σ∇f, ∇f ⟩] , and Var Q (f 2 ) ≤ E Q [⟨Σ∇f, ∇f ⟩] . Thus, Q is L-Sob(c LS ) with constant c LS (Q) = 2∥Σ∥ op and also Poinc(c LS ) with constant c LS (Q) = ∥Σ∥ op
, where ∥.∥ op denotes the operator norm.

In Proposition 2, the first inequality can be derived from the classical log-Sobolev inequality for N (0, Id) stated in [START_REF] Gross | Logarithmic Sobolev Inequalities[END_REF], with a change of variable. Similarly, the Poincaré inequality can be obtained through a change of variable from the Poincaré inequality for N (0, Id) which is a particular case of the Brascamp-Lieb inequality for log-concave probability measures [START_REF] Herm | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] and is stated explicitly in Beckner (1989, Theorem 1). We now focus on specific posterior distributions called Gibbs posteriors, or Gibbs distributions. For a fixed loss ℓ and dataset S m , the Gibbs posterior, w.r.t. prior P ∈ M(H), risk RSm and inverse temperature γ > 0.is defined as P -γ RSm such that dP -γ RSm (h) ∝ exp(-γ RSm (h))dP (h). Gibbs posteriors are a class of closed-form solutions for relaxation of Catoni (2007, Theorem 1.2.6) stated, for instance, in Alquier et al. (2016, Theorem 4.1). Proposition 3 shows that when the prior and the loss satisfies a few properties, then the associated Gibbs posterior is L-Sob(c LS ).

Proposition 3 Assume that P is a probability measure on R d such that dP (h) ∝ exp(-V (x)) with V a smooth function such that Hess(V ) ⪰ 2 c LS (P) Id. Assume that ℓ = ℓ 1 + ℓ 2 with ℓ 1 convex, twice differentiable and ℓ 2 bounded. Then for any γ > 0, the Gibbs posterior

Q = P -γ RSm is L-Sob(c LS ) with constant c LS (Q) = c LS (P) exp (4∥ℓ 2 ∥ ∞ ).
Proposition 3 applies, e.g., when P is a Gaussian prior P = N (µ P , Σ P ). Notice that in this case c LS (P) = 2∥Σ P ∥ op . This property is a straightforward application of Chafaï (2004, Corollary 2.1) with Guionnet and Zegarlinksi (2003, Property 2.6) and is stated in Appendix A for completeness. Finally, notice that satisfying a log-Sobolev inequality is stronger than satisfying a Poincaré one. This is stated for instance in Ledoux (2006, Proposition 2.1) and properly recalled in Appendix A.

3. Reaching a flat minimum allows Poincaré posteriors to generalise well 3.1. Fast rate PAC-Bayes bounds for heavy-tailed losses

In order to obtain fast rates, i.e., bounds converging to zero faster than 1 / √ m, we exploit the notion of flat minimum (where the loss takes a small value in the neighbourhood of the minimum). Indeed, in an overparametrised setting such as neural networks, it is likely to obtain such a minimum once the optimisation phase has been performed, as there are much more parameters than training data. We exploit this flatness property within PAC-Bayes bounds through the gradient norm ∥∇ h ℓ(., z)∥ of the loss w.r.t. the predictor h for any z. This is, to the best of our knowledge, the first attempt to do so as [START_REF] Gat | On the Importance of Gradient Norm in PAC-Bayesian Bounds[END_REF] focus on gradients with respect to the data ∇ z ℓ (one does not optimise on those, as the dataset is fixed in practice).

In this section, we consider posterior distributions Q being Poinc(c P ). This assumption covers the important case of Gaussian measures (Proposition 2) as well as all measures satisfying a log-Sobolev inequality (Proposition 14). We focus on PAC-Bayes bound holding for distributions Q satisfying a particular assumption involving the data distribution D (contrary to many PAC-Bayes bounds holding for all Q). We then define the error of Q ∈ M(H) for any datum z ∈ Z as Err(ℓ, Q, z) := E h∼Q [ℓ(h, z)] and identify Assumption 4 to later involve flat minima.

Assumption 4 We say that Q ∈ M(H) is quadratically self-bounded with respect to ℓ and constant

C > 0 (namely QSB(ℓ, C)) if E z∼D Err(ℓ, Q, z) 2 ≤ CR D (Q) (= CE z∼D [Err(ℓ, Q, z)]) Assumption 4 is a relaxation of boundedness, as if ℓ ∈ [0, C] then it is QSB(ℓ, C).
It is an alternative to the bounded expected variance assumption in anytime-valid PAC-Bayes bounds (Haddouche and Guedj, 2023a;[START_REF] Chugg | A Unified Recipe for Deriving (Time-Uniform) PAC-Bayes Bounds[END_REF]). An issue with such boundedness assumption is that it has to hold for all posteriors, including those providing poor generalisation performances. This is avoided by the QSB assumption which intricate the properties of D, ℓ and Q. Such a design is in line with the conclusions of the recent work of [START_REF] Gastpar | Fantastic Generalization Measures are Nowhere to be Found[END_REF], inviting to derive generalisation bounds valid for specific pairs (Q, D) (and not uniformly valid for all such pairs). Finally, we interpret C as a contraction constant attenuating, on average, the local expansion (governed by variances of Q, and D) of the loss around the mean of Q. Exploiting the PAC-Bayes supermartingales bounds of Haddouche and Guedj (2023a); [START_REF] Chugg | A Unified Recipe for Deriving (Time-Uniform) PAC-Bayes Bounds[END_REF] alongside Poincaré inequality leads to the following.

Theorem 5 For any C > 0, any 2 C > λ > 0, any data-free prior P, any ℓ ≥ 0 and any δ ∈ [0, 1], we have, with probability at least 1 -δ over the sample S, for any m > 0, any Q being Poinc(c P ), QSB(ℓ, C) and ℓ(.,

z) ∈ H 1 (Q) for all z, R D (Q) ≤ 1 1 -λC 2 RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 -λC c P (Q) E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 .
This theorem shows that, for any posterior being QSB w.r.t. the distribution D, fast rates are achievable as long as RSm ≈ 0, and expected gradients are vanishing. While the first condition is often involved for deep neural networks in the overparametrised setting, the second holds if a flat minimum has been reached through the optimisation process. Then, taking λ = 1 /C ensures an anytimevalid PAC-Bayesian bound with a fast rate of 1 /m. Otherwise, for a fixed m, taking λ = m -α /C, α ∈ [0; 1 /2] allows to adapt the convergence speed w.r.t. the behaviour of the gradients. In the case of constant gradients, we recover a convergence rate of 1 / √ m, matching Alquier et al. (2016, Theorem 4.1). On the role of flat minima in PAC-Bayes learning. Theorem 5 suggests that, in order to attain good generalisation ability, the mean of Q has to be close from two minima: (i) on RSm in order to make RSm small, and (ii) on E z∼D [∥∇ h ℓ(h, z)∥ 2 ] to make the gradients small. The variance of Q has to fit the flatness of those minima, the flatter they are, the larger the variance in order to shrink the expected terms on the right-hand-side of Theorem 5. Finally, the KL term invites, e.g. for Gaussian distributions, to consider high variances, hence flat minima to maintain a small value of the bound. A focus on C. Taking λ = 1 /C in Theorem 5 attenuates the impact of the prior distribution and amplifies the gradient term. Then, a small C is desirable when working with flat minima to attenuate an ill-designed prior. Having a small C is reachable in practice: we show in Section 6, for a classification task on MNIST, that the QSB assumption is verified with C strictly smaller than 1 when considering neural networks. High probability bounds with fast rates, a paradox? Grunwald et al. (2021, page 7) showed that, for a trivial H = {h} ⊂ R d , for any loss, any i.i.d. dataset S m with variance σ 2 , we have asymptotically, with probability at least α, for a constant C α depending on α and N (0, Id), we have

R D (h) ≥ RSm (h) + C α σ 2 √ m .
Is it paradoxical with Theorem 5? The answer is no: the bound in [START_REF] Grunwald | PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes[END_REF] gives an asymptotic lower bound on the convergence of

RSm (h) to R D (h). Theorem 5 informs us on how R D is getting closer from 1 1-λ/2 RSm which converges to 1 1-λ/2 R D > R D
as the loss is non-negative. Theorem 5 then show the existence of a 'transition regime' involving a fast rate. Once 1 1-λ/2 RSm is reached, the clower bound of [START_REF] Grunwald | PAC-Bayes, MAC-Bayes and Conditional Mutual Information: Fast rate bounds that handle general VC classes[END_REF] ensures an asymptotic regime with slow convergence rate. Note that such transition regimes already appeared in the literature in Tolstikhin and Seldin (2013); Mhammedi et al. (2019) at the cost of additional variance terms compared to Theorem 5. However, such fast rates have never been linked before to flat minima (and optimisation in general), highlighting the potential of our bound to explain the ability of deep neural networks to generalise well in the overparametrised setting (m far smaller than the dimension of H), where flat minima are likely to be reached, as studied, e.g., in [START_REF] Karolina Dziugaite | In search of robust measures of generalization[END_REF], showing correlations between flat minima and generalisation for various learning problems.

Proof [Proof of Theorem 5] We start from Chugg et al. (2023, Corollary 17) instantiated with a single λ, i.i.d. data and a prior P. With probability at least 1 -δ, for any Q ∈ M(H) and m > 0:

R D (Q) ≤ RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 E h∼Q E z∼D [ℓ(h, z) 2 ] ,
where z ∼ D is independent from S. We study the last term on the right-hand side. First, applying Fubini's theorem gives:

E h∼Q E z∼D [ℓ(h, z) 2 ] = E z∼D E h∼Q [ℓ(h, z) 2 ] = E z∼D Var h∼Q (ℓ(h, z)) + E h∼Q [ℓ(h, z)] 2 .
As for any z, ℓ(., z) ∈ H 1 , we apply Poincaré's inequality to obtain:

≤ E z∼D c P (Q) E h∼Q ∥∇ h ℓ(h, z)∥ 2 + E h∼Q [ℓ(h, z)] 2 .
Using that Q is QSB(ℓ, C) and re-organising the terms gives:

R D (Q) ≤ 1 1 -λC 2 RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 -λC c P (Q) E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 .
It is possible to go beyond the QSB assumption. This comes at the cost of an upper bound on R D as well as a supplementary Poincaré assumption on D.

Corollary 6 For any C > 0, any δ ∈ (0, 1) any 2 C > λ > 0, any data-free prior P, any ℓ ≥ 0 such that, for any z ∈ Z, we have ℓ(., z) ∈ H 1 and for any h, the loss function ℓ(h, .) is C 1 almost everywhere on Z. If the data distribution D is Poinc(c P ), then with probability at least 1 -δ over the sample S, for any m > 0, any posterior

Q being Poinc(c P ) with R D (Q) ≤ C: R D (Q) ≤ 1 1 -λC 2 RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 -λC c P (Q) E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 + c P (D) E z∼D E h∼Q [∇ z ℓ(h, z)] 2 .
Proof is deferred to Section C.1. Corollary 6 states that, if Q reached a flat minimum (meaning ∥∇ h ℓ∥ is small), and this minimum is robust to the training dataset (meaning ∥∇ z ℓ∥ is small), then a fast rate is attainable while only requiring an upper bound on R D (Q). This conclusion holds when D Poinc, encompassing the case of Gaussian mixtures [START_REF] Schlichting | Poincaré and Log-Sobolev Inequalities for Mixtures[END_REF], which can approximate any smooth density (as recalled in [START_REF] Gat | On the Importance of Gradient Norm in PAC-Bayesian Bounds[END_REF]. However, the Poincaré constant of a general mixture is not known, and the upper bound of [START_REF] Schlichting | Poincaré and Log-Sobolev Inequalities for Mixtures[END_REF] scales with the number of components, involving potentially high χ 2 divergences.

Comparison with [START_REF] Gat | On the Importance of Gradient Norm in PAC-Bayesian Bounds[END_REF]. We compare Corollary 6 with Gat et al. (2022, Theorems 3.5, 3.6). First, our result holds with the assumption that D follows a Poincaré inequality, which is strictly less restrictive than assuming a log-Sobolev inequality (Proposition 14). Second, they assume a bounded loss and their result holds only for classification problem satisfying a technical assumption on the label repartition (see their Lemma 3.3) while ours holds for any learning problem at the sole assumption of a bounded R D (Q), allowing ℓ to be non-negative. Moreover, note that to conclude their proof, [START_REF] Gat | On the Importance of Gradient Norm in PAC-Bayesian Bounds[END_REF] had to use a uniform bound on Ez[∥∇ z ℓ∥] in their Theorem 3.5 to have a tractable bound, thus the benefits of gradient norm is unclear. While they overcome this limitation in Gat et al. (2022, Theorem 3.6), the explicit influence of the gradient norm appears within an exponential moment on the losses (attenuated by a logarithm). However, a major limitation is that this exponential moment is averaged w.r.t. P, being data-free. Thus, the associated gradients have no apparent reason to be small, and their result cannot be linked to flat minima, contrary to Corollary 6 involving expected gradients w.r.t. Q, being the output of an optimisation process.

3.2. Towards fully empirical bound for gradient-Lipschitz functions.

In this section, we assume the loss ℓ is such that, for any z ∈ Z, the gradient Theorem 7 For any C 1 , C 2 , c > 0, any data-free prior P, any ℓ ≥ 0 being C 2 and any δ ∈ [0, 1], we have, with probability at least 1 -δ over the sample S, for any m > 0, any Q being Poinc(c P ) with constant c, QSB(ℓ, C 1 ), QSB ∥∇ h ℓ∥ 2 , C 2 and ℓ(., z),

∇ h ℓ(., z) is G- Lipschitz,
∥∇ h ℓ∥ 2 (., z) ∈ H 1 (Q) for all z, R D (Q) ≤ 2 RSm (Q) + 2c C 1 E h∼Q 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 + 2 C 1 + c 4cG 2 + C 2 C 1 KL(Q, P) + log(2/δ) m .
Proof is deferred to Section C.2. Here, we showed that to attain fast rates, the QSB assumption has to be reached for both the loss and its gradient. This suggests several things on the flat minimum that has to be reached by Q (designed from RS ): first, it needs to be close from a flat minimum of R D to satisfy the QSB assumption. Second, this minimum also ensures the contraction of the gradients. We then are able to derive an empirical generalisation bound, involving both empirical loss and gradients. Not only Theorem 7 yields, to our knowledge, the first PAC-Bayesian algorithm involving gradient terms, but also can be translated to a generalisation metric in order to understand generalisation. Such an idea has been exploited recently [START_REF] Behnam Neyshabur | Exploring Generalization in Deep Learning[END_REF][START_REF] Jiang | Fantastic Generalization Measures and Where to Find Them[END_REF][START_REF] Karolina Dziugaite | In search of robust measures of generalization[END_REF]. In particular, from RS (Q), Neyshabur et al. ( 2017) derived a notion of sharpness, stated in Equation ( 1), aiming to be informative on the flatness of the reached minima for any Q = N (µ Q , σ 2 Id). This notion is defined by

E ν∼N (0,σ 2 Id) RSm (µ Q + ν) -RSm (µ Q ) . (1)
Theorem 7 enhance this notion of sharpness by involving the empirical gradients when Q is QSB(ℓ, C 1 ):

Sharp σ 2 C 1 (Q) := E ν∼N (0,σ 2 Id) 2 RSm + σ 2 C 1 G-RSm (µ Q + ν) -2 RSm + σ 2 C 1 G-RSm (µ Q ) ,
(2) where G-

RSm (h) = 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 .
This gradient term can be seen as an empirical Fisher information, linked to the second-order moment derivative. Thus, (2) involves a notion of flatness on both the loss and its gradient, contrary to (1). For the sake of clarity, we particularise Theorem 7 in Corollary 8 with Gaussian distributions and this novel notion of sharpness.

Corollary 8 For any C 1 , C 2 > 0, any fixed variance σ 2 > 0, any data-free prior P = N (µ P , σ 2 Id), any nonnegative loss ℓ being C 2 and any δ ∈ [0, 1], we have, with probability at least 1 -δ over the sample S, for any m > 0, any

Q = N (µ Q , σ 2 Id) being QSB(ℓ, C 1 ), QSB ∥∇ h ℓ∥ 2 , C 2 and ℓ(., z), ∥∇ h ℓ∥ 2 (., z) ∈ H 1 (Q) for all z, R D (Q) ≤2 RSm (µ Q ) + G-RSm (µ Q ) + Sharp σ 2 C 1 (Q) + O KL(Q, P) + log(2/δ) m .

Generalisation ability of Gibbs distributions with a log-Sobolev prior

One limitation of the results given in Section 3 is that the KL divergence term remains uncontrolled in general as its formulation depends on the nature of P and Q. A close form exists for Gaussian distributions for instance, but this class of distribution is limiting. Perpetrating the spirit of [START_REF] Catoni | PAC-Bayesian supervised classification: the thermodynamics of statistical learning[END_REF], we go beyond the Gaussian distributions to focus on the Gibbs posteriors which have naturally appeared in PAC-Bayes through the use of tools from statistical physics. We show that log-Sobolev inequalities allow us to control the KL divergence of such distributions w.r.t. their priors.

Controlling the KL divergence when Q is a Gibbs posterior. Lemma 9 exploits the fact that the KL divergence can be formulated as an entropy w.r.t. the prior distribution P. It then shows that the KL divergence of the Gibbs posterior P -γ RSm w.r.t. P is upper bounded by gradient terms as long as P satisfies a log-Sobolev inequality.

Lemma 9 For any m, P being L-Sob(c LS ), any ℓ ≥ 0 such that for any z, ℓ(., z) ∈ H 1 (P), we have, for any γ > 0:

KL P -γ RSm , P ≤ γ 2 c LS (P) 4 E h∼P -γ RSm ∥∇ h RSm (h)∥ 2 .
Proof is deferred to Section C.3. The crucial message of this lemma is that, a flat minimum of RS allows controlling the KL divergence. This message is new and independent of Section 3 which focus on flat minima reached for R D . Note that in this case, the KL divergence has an explicit formulation. However it involves to calculate the exponential moment E h∼P [exp(-γ RSm )] which is costly in practice. On the contrary, we only need to estimate a second-order moment over P -γ RSm . Generalisation ability of Gibbs posteriors. When Gibbs posteriors are involved, KL divergence is controllable by a gradient term. An ideal way to conclude would be, as in Section 3 to involve Poincaré inequality. However, Gibbs posterior are not necessarily satisfying a Poincaré inequality as in Section 3, we then need to make supplementary assumptions on the loss.

Theorem 10 For any C > 0, any γ > 0, any prior P being L-Sob(c LS ), any ℓ ≥ 0 and any δ ∈ [0, 1], we have the following inequalities. If ℓ ∈ [0, 1], then with probability at least 1 -δ over the sample S, for any m > 0, and any Q ∈ M(H):

R D (P -γ RSm ) ≤ 2 RSm (P -γ RSm ) + γ 2 c LS (P) 4m E h∼P -γ RSm ∥∇ h RSm (h)∥ 2 + log(1/δ) m .
If ℓ = ℓ 1 + ℓ 2 with ℓ 1 convex, twice differentiable and ℓ 2 bounded, assume that P satisfies the conditions of Proposition 3. Then for any 2 C > λ > 0, with probability at least 1 -δ over the sample S, for any m > 0, such that Q is QSB(ℓ, C) and ℓ(., z) ∈ H 1 (P -γ RSm ):

R D (P -γ RSm ) ≤ 1 1 -λC 2 RSm (P -γ RSm ) + γ 2 c LS (P) 4λm E h∼P -γ RSm ∥∇ h RSm (h)∥ 2 + log(1/δ) λm + λe 4∥ℓ 2 ∥∞ c LS (P) 4 -2λC E z∼D E h∼P -γ RSm ∥∇ h ℓ(h, z)∥ 2 .
Proof is deferred to Section C.4. Note that we could have derived analogous to Corollary 6 at the cost of a supplementary Poincaré assumption on D. The influence of the inverse temperature γ is quadratic: this is the price to pay to fit the dataset and reduce the influence of the prior. This dependency is therefore attenuated by a gradient term, small if a flat minimum on the empirical risk has been reached. This suggests that in the case of Gibbs posteriors with log-Sobolev prior, reaching a flat minima on RSm controls not only RSm (Q), but also the KL divergence and this last point is not reachable when considering Poincaré distributions. The other gradient term comes from Section 3 and requires to be close from a flat minimum on R D to attain fast rates.

On the benefits of the gradient norm in Wasserstein PAC-Bayes learning

In Sections 3 and 4, we provided various generalisation bounds, benefiting from flat minima. However, our results involve a KL divergence, implying absolute continuity of Q w.r.t. P, incompatible with the case of deterministic predictors (Dirac distributions). To circumvent this issue, a recent line of work emerged, involving integral probability metrics, with a particular focus on the 1-Wasserstein distance Amit et al. ( 2022); Haddouche and Guedj (2023b); Viallard et al. (2023b). The idea behind these works is to replace the change of measure inequality [START_REF] Csiszar | I-Divergence Geometry of Probability Distributions and Minimization Problems[END_REF][START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time-III[END_REF]) by the Kantorovich-Rubinstein duality [START_REF] Villani | Optimal transport: old and new[END_REF] to trade a KL for a Wasserstein. We go even further here by obtaining the first PAC-Bayesian bound involving directly a 2-Wasserstein distance (see definition 15), trading Lipschitz assumption for gradient-Lipschitz one (well-suited for optimisation). To do so, we first derive a novel change of measure inequality.

Theorem 11 Assume H to have a finite diameter D > 0. Then for any function f : H → R with G-Lipschitz gradients, the following holds: for all distributions P, Q ∈ M(H) 2 ,

E h∼Q [f (h)] ≤ G 2 W 2 2 (Q, P ) + E h∼P [f (h)] + DE h∼Q [∥∇f (h)∥].
Proof is deferred to Section C.5 Theorem 11 shows it is possible when gradients are Lipschitz, to obtain a duality formula involving the gradient of the considered function at the price of a linear dependency on the diameter of H. Theorem 11 is also linked to the change of measure inequality when the prior distribution satisfies a log-Sobolev inequality.

Corollary 12 Assume that P is such that dP ∝ exp(-V )dx with V being C 2 and P is L-Sob(c LS ).

Then, for any R > 0, any f with gradients G-Lipschitz on B(0, R), and any distributions P, Q,

E h∼Q [f (P R (h))] ≤ Gc LS (P) 4 KL(Q, P) + E h∼P [f (P R (h))] + 2RE h∼Q [∥∇f (P R (h))∥] ,
where P R denotes the Euclidean projection on B(0, R).

Proof is deferred to Section C.6. Corollary 12 involves a KL divergence and an Euclidean predictor space H = R d . This comes at the cost of approximating Q, P by P R #Q, P R #P. Thus, R is now an hyperparameter which arbitrates a tradeoff between the quality of our approximations and the looseness of the bound (if the gradient norm is large). A notable strength is that the smoothness assumption is relaxed on smoothness over B(0, R).

From Theorem 11, we now derive a novel generalisation bound allowing deterministic predictors.

Theorem 13 Let δ ∈ (0, 1) and P ∈ M(H) a data-free prior. Assume H has a finite diameter D > 0, ℓ ≥ 0 and that for any m, the generalisation gap ∆ Sm is G gradient-Lipschitz. Assume that E h∼P E z∼D [ℓ(h, z) 2 ] ≤ σ 2 , then the following holds with probability at least 1 -δ, for any m > 0 and any Q:

R D (Q) ≤ RSm (Q) + G 2 W 2 2 (Q, P) + 2σ 2 log 1 δ m + DE h∼Q ∇ h R D (h) -∇ h RSm (h)
Proof is deferred to Section C.7. Theorem 13 is not the first generalisation bound to involve a 2-Wasserstein distance (Lugosi andNeu, 2022, 2023). However, those results involve infinitely smooth loss functions. Also, results from Amit et al. ( 2022); Haddouche and Guedj (2023b); Viallard et al. ( 2023b) using 1-Wasserstein can be directly relaxed on bounds involving the 2-Wasserstein, while still requiring a Lipschitz loss. On the contrary, our result holds for any nonnegative gradient-Lipschitz ∆ Sm , which is well-suited for optimisation. Theorem 13 involves a slow rate of 1 / √ m as we have to control the generalisation gap w.r.t. to P. It is possible to make appear the gradients expected over P using the QSB assumption, but we have no reason to expect those gradients to be small, we then controlled this term uniformly by σ 2 . Another restriction of our result compared to previous ones is that it holds for H having a finite diameter, however, having a small expected ∥∇ h R D -∇ h RSm ∥ over Q (which is the case when flat minima on both empirical and true risks are reached) allows taking D large, and thus, having good approximations of measures on a Euclidean space through orthogonal projections as in Corollary 12.

An empirical study of Assumption 4 for neural networks

In this section, we check empirically whether the QSB assumption is verified for neural networks. This allows us to verify if Theorem 5 is useful to understand the generalisation ability of neural nets.

Experimental protocol. We consider classification tasks on two datasets: MNIST [START_REF] Lecun | The MNIST database of handwritten digits[END_REF] and FashionMNIST [START_REF] Xiao | Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms[END_REF]. We kept the original training set S m and the original test set denoted by T n (of size n). We consider the convolutional neural network of [START_REF] Tobias Springenberg | Striving for simplicity: The all convolutional net[END_REF] adapted for MNIST and FashionMNIST. The model is composed of 4 layers containing 10 channels with a 5×5-kernel; we set the stride and the padding to 1, except for the second layer, where it is fixed to 2. Each of these (convolutional) layers is followed by a Leaky ReLU activation function. Moreover, an average pooling with a 8×8-kernel is performed before the Softmax activation function. To initialise the weights of the network, we use [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF] uniform initializer, while the biases are initialised in [-1 / √ 250, + 1 /1/ √ 250] uniformly (except the first layer, the interval is [-1 /5, + 1 /5]). Hence, in this case, H is the set of neural networks with a fixed architecture, and parametrised with a vector w. while the posterior distribution Q is a Gaussian measure N (w, σ 2 Id) centered on the parameters w associated with the model; σ is set to 10 -4 . Note that this distribution respects the Poinc(c P ) assumption; see Section 3.1. We train the neural network with the (vanilla) stochastic gradient descent algorithm, where the batch size is equal to 512, and the learning rate is fixed to 10 -2 . We train for at least 10 4 gradient steps and finish the current epoch when this number of iterations is reached. Our loss ℓ is the bounded cross-entropy loss of Dziugaite and Roy (2017, Section D).

In Figure 1, we report the evolution of three quantities: (i) the estimated value of C, (ii) the test risk RTn (Q) and (iii) the test risk with the 01-loss. More precisely, for computational reasons, the risks and C are estimated by sampling one hypothesis h ∼ Q and by computing the values on a mini-batch of T n (with 512 examples) at each iteration. Then, Figure 1 represents averaged values on 5 runs, each point of the curve representing the average on 100 iterations of the training process (for 10 4 iterations we only plot 10 2 averaged points for clarity).

Empirical findings. Figure 1 illustrates that, when neural networks are involved for two classification tasks, Q evolves during the optimisation process while maintaining the QSB property with constant C < 1. For both MNIST and FashionMNIST, the constant C decreases from approximately 0.55 to 0.45. We deduce two things from this: (i) the learning phase, while optimising RSm also gain in generalisation ability, shrinking the averaged loss on new data which is translated by a smaller C; and (ii), having a data-free P (0 iteration) being QSB with C < 1 suggests that the architecture of our neural network also has an influence on the QSB assumption. As precised in Section 3, having C < 1 attenuates the impact of the KL term, thus P. This is desirable as it allows the optimiser to deeply explore the predictor space when P yields poor performances. We also note that the generalisation ability of Q on the training loss nearly matches the performance on the 0-1 loss for MNIST but is deteriorated for FashionMNIST, this invites to study more deeply the design of such surrogates in future work. Finally, the take-home message of this study is that the QSB assumption is verified for neural networks on MNIST. Such an empirical confirmation is crucial as it is required for our main result (Theorem 5) and thus confirms that, for neural networks, reaching flat minima during the optimisation phase translates in increased generalisation ability.

Conclusion

We provide novel PAC-Bayes generalisation bounds, converging faster than 1 / √ m when a low empirical error is reached and that expected gradients are vanishing. This conveys the message that flat minima helps generalisation. However, to complete this analysis, the crucial question is to understand how optimisation algorithms successfully reach flat minima in the overparametrised setting. This important question is left as future work.

A.2. Wasserstein distances

We recall here the definition of Wasserstein distances, valid for any Polish space H equipped with a distance d.

Definition 15 The 1-Wasserstein distance between P, Q ∈ M(H) 2 is defined as

W 1 (Q, P ) = inf π∈Π(Q,P ) H 2 ||x -y||dπ(x, y).
where Π(Q, P ) denote the set of probability measures on H 2 whose marginals are Q and P . We define the 2-Wasserstein distance on P(H) as Corollary 16 Let λ > 0, m ≥ 1 and a data-free prior P. Assume that for any h ∈ H, ℓ(h, .) is L-Lipschitz and that the data distribution D is L-Sob(c LS ). Then for with probability at least 1 -δ over S, for any Q ∈ M(H),

W 2 (Q, P ) = inf π∈Π(Q,P ) H 2 ||x -y|| 2 dπ(x,
R D (Q) ≤ RSm (Q) + KL(Q, P) + log(1/δ) λ + 2λ 2 L 2 c LS (D) m .
Proof We take f (h) = λ∆ S (h) := λ(R D (Q)-RSm (Q)) first use the change of measure inequality [START_REF] Csiszar | I-Divergence Geometry of Probability Distributions and Minimization Problems[END_REF][START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time-III[END_REF] to state that, for any Q,

E h∼Q [f (h)] ≤ KL(Q, P) + log (E h∼P [exp (f (h))]
) . Markov's inequality alongside Fubini's theorem gives, with probability at least 1 -δ, (Ané et al., 2000, Corollary 3.2.3). Then, using Herbst argument similarly as in Ledoux (2006, Section 2.3) allow us to conclude that f is 2L c LS (D)-subgaussian, thus,

E h∼Q [f (h)] ≤ KL(Q, P) + log(1/δ) + log (E h∼P E S [exp (f (h))]) . Now, we use that f is L-Lipschitz for all h, then the function S → ∆ S (h) is 2 √ m -Lipschitz on S for each h. As D is L-Sob(c LS ) inequality, D ⊗m is also L-Sob(c LS ) with identical constant
log (E h∼P E S [exp (f (h))]) ≤ 2λ 2 L 2 m .
This concludes the proof.

Disintegrated PAC-Bayes bounds Numerical estimation of PAC-Bayes bounds is usually challenging as it often involves Monte-Carlo approximations of the expectation over the posterior Q. A recent line of work (Rivasplata et al., 2020;[START_REF] Haddouche | Online PAC-Bayes Learning[END_REF]Viallard et al., 2023a) studies disintegrated PAC-Bayes bounds e.g., bounds holding with high-probability on both the dataset S and a single predictor h drawn from the posterior Q. Those bounds are relevant for practitioners as they require little computational time. However, a drawback of these bounds is that existing disintegrated bounds do not allow the KL divergence to be used as a complexity measure. Either disintegrated KL (Rivasplata et al., 2020) or Rényi divergences (Viallard et al., 2023a), which can be seen as a relaxation of the KL one, are considered.

Using again the subgaussianity behavior of Lipschitz losses, it is possible to attain PAC-Bayesian disintegrated bounds as long as the posterior distribution satisfies a log-Sobolev inequality with sharp constant (achievable for instance for Gaussian distribution with small operator norm).

Lemma 17 Assume that for any z, ℓ(., z) is L-Lipschitz and that Q is Poinc(c P ) with c P (Q) ≤ 1/m. Then, with probability 1 -δ over the draw of h ∼ Q:

∆ Sm (h) ≤ ∆ Sm (Q) + 2L 2 log(1/δ) m .
This lemma states that, as long as we assume our loss to be Lipschitz w.r.t. h, then it is possible to easily derive disintegrated PAC-Bayesian bounds. Also notice that Lemma 17 is easily completed by Corollary 16 which makes appear a KL divergence as complexity. Note also that as the loss is Lipschitz, it is also possible to make appear 1-Wasserstein distance through the bounds of Haddouche and Guedj (2023b); Viallard et al. (2023b). Thus Having a Log-Sobolev assumption with sharp constant on the posterior distribution is enough to provide disintegrated PAC-Bayesian bounds involving KL or Wasserstein terms instead of Rényi divegerences or disintegrated KL.

Appendix C. Proofs

C.1. Proof of Corollary 6

To start this proof, we first state an important intermediary theorem, holding at no assumption on the data-distribution.

Theorem 18 For any C > 0, any 2 C > λ > 0, any data-free prior P, any nonnegative loss function ℓ such that, for any z ∈ Z, ℓ(., z) ∈ H 1 , and any δ ∈ [0, 1], we have, with probability at least 1 -δ over the sample S, for any m > 0, any posterior Q being Poinc(c P ), such that R D (Q) ≤ C and such that for any z, ℓ(.,

z) ∈ H 1 (Q): R D (Q) ≤ 1 1 -λC 2 RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 -λC c P (Q) E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 + Var z∼D E h∼Q [ℓ(h, z)] .
Theorem 18 exhibits the influence of the gradient norm of ∇ h ℓ on the generalisation ability: small gradients makes the bound vanish, the remaining variance term is not treated for now and can be assumed bounded, but we cannot then recover a fast rate. We show next that assuming additional assumption over the data distribution circumvent this issue. Proof We re-start from Chugg et al. (2023, Corollary 17), for any λ > 0, with probability at least 1 -δ, for any m > 0, any posterior Q:

R D (Q) ≤ RSm (Q) + KL(Q, P) + log(1/δ) λm + λ 2 E h∼Q E z∼D [ℓ(h, z) 2 ] .
Then, the last term is controlled as follows,

E h∼Q E z∼D [ℓ(h, z) 2 ] ≤ E z∼D c P (Q) E h∼Q ∥∇ h ℓ(h, z)∥ 2 + E h∼Q [ℓ(h, z)] 2 .
We then make appear a supplementary variance term:

= E z∼D c P (Q) E h∼Q ∥∇ h ℓ(h, z)∥ 2 + Var z∼D E h∼Q [ℓ(h, z)] + E z∼D E h∼Q [ℓ(h, z)] 2 .
Note that by Fubini, the last term on the right-hand side is exactly R D (Q) 2 , then using that the averaged true risk is lesser than C, and re-organising the terms in Chugg et al. (2023, Corollary 17) gives, for λ ∈ 0, 2 C :

R D (Q) ≤ 1 1 -λC 2 RSm (Q) + KL(Q, P) + log(1/δ) λ 1 -λC 2 m + λ 2 -λC c P (Q) E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 + Var z∼D E h∼Q [ℓ(h, z)] .
Now Theorem 18 is proven, we only need to exploit the Poincaré assumption on the data distribution on the variance term to obtain Corollary 6.

C.2. Proof of Theorem 7

Proof We start again from Theorem 5, with λ = 1/C 1 then have with probability 1 -δ/2:

R D (Q) ≤ 2 RSm (Q) + 2C 1 KL(Q, P) + log(2/δ) m + c P (Q) C 1 E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 .
(3)

We now remark that g(h, z) := ∥∇ h ℓ(h, z)∥ 2 is nonnegative. Then, given our assumptions, we apply the route of proof of Theorem 5 on g i.e. we start again from the (Chugg et al., 2023, Corollary 17), apply Poincaré's inequality on Q and use the QSB assumption on g. We then have for any λ > 0, with probability at least 1 -δ/2, any Q being Poinc(c P ), QSB (g, C 2 ) and g(., z) ∈ H 1 (Q) for all z :

E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 ≤ E h∼Q 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 + KL(Q, P) + log(2/δ) λm + λc P (Q) 2 E z∼D E h∼Q ∥∇ h g(h, z)∥ 2 + λC 2 2 E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 . (4)
Finally, notice that, by definition of g, ∇ h g(h, z) = 2Hess h (ℓ)(h, z)∇ h ℓ(h, z), where Hess h (ℓ) denotes the Hessian of ℓ. Thus, using that ℓ(., z) is G gradient Lipschitz for any z gives, for any (h, z) that ∥∇ h g(h, z)∥ ≤ 2G∥∇ h ℓ(h, z)∥. Plugging this in (4) gives:

E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 ≤ E h∼Q 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 + KL(Q, P) + log(2/δ) λm + λ 2 4c P (Q)G 2 + C 2 E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 . (5)
Finally, using that c P (Q) = c, taking λ = 1 4cG 2 +C 2 and re-organising the terms in (5) gives:

E z∼D E h∼Q ∥∇ h ℓ(h, z)∥ 2 ≤ 2 E h∼Q 1 m m i=1 ∥∇ h ℓ(h, z i )∥ 2 + 2(4cG 2 + C 2 ) KL(Q, P) + log(2/δ) m (6)
Finally, taking an union bound and plugging (6) in (3) concludes the proof.

C.3. Proof of Lemma 9

Proof For conciseness, we rename Q := P -γ RSm . We first notice that, denoting by dQ dP the Radon-Nikodym derivative of Q with respect to P:

KL P -γ RSm , P = E h∼Q log dQ dP (h) = Ent P dQ dP = Ent P [g 2 ],
where g = dQ dP . Recall that dQ dP (h) = 1 Z exp -γ RSm (h) where Z = E h∼P exp -γ RSm (h) . Then, g(h) = 1 √ Z exp(-γ 2 RSm (h)) belongs in H 1 (P) as long as ℓ ∈ H 1 . Indeed, as exp is infinitely smooth, g ∈ D 1 (R d ), also as the loss is nonnegative, then g ≤ 1 √ Z thus g ∈ L 2 (P).

Finally, ∇g = -γ 2 g(h)∇ RSm (h). As g(h) ≤ 1 √ K , we only need to bound ∥∇ RSm (h)∥ 2 to ensure that g ∈ H 1 (P):

∥∇ RSm (h)∥ 2 = 1 m 2 1≤i,j≤m ⟨∇ℓ(h, z i ), ∇ℓ(h, z j )⟩ ≤ 1 2m 2
1≤i,j≤m ∥∇ℓ(h, z i )∥ 2 + ∥∇ℓ(h, z j )∥ 2

As we assumed ∥∇ℓ(., z)∥ 2 ∈ L 2 (P) for all z, we conclude that g ∈ H 1 .

We then can apply the log-Sobolev inequality to conlude that where z ∼ D is independent of S.

For the first inequality, we just take λ = 1, we use that ℓ(h, z) 2 ≤ ℓ(h, z) and re-organise the terms. Finally, we upper bound the KL term thanks to Lemma 9.

For the second inequality, we exploit Proposition 3 to use the fact that P -γ RSm is L-Sob(c LS )

alongside Proposition 14 which ensures that P -γ RSm is Poinc(c P ) with constant equal to c LS P -γ RSm /2. We then apply a route of proof similar to Theorem 5. We have : Finally, using that P -γ RSm is QSB(ℓ, C) allow us to re-organise the terms as in Theorem 5. Combining this with Lemma 9 to bound the KL divergence concludes the proof.

E h∼P -γ RSm

C.5. Proof of Theorem 11

Proof We assume first G = 1. We start from the Kantorovich duality formula (Villani, 2009, Theorem 5.10) instantiated with the cost function c(x, y) = ∥x -y∥ 2 . We have for any Q, P, because W 2 is a distance:

W 2 (Q, P) = W 2 (P, Q) = sup ϕ,ψ E h∼Q [ϕ(h)] -E h∼P [ψ(h)], (7)
where the supremum is taken over the functions ϕ, ψ ∈ L 1 (Q) × L 1 (P) such that for all h, h ′ ∈ H 2 , ϕ(h) -ψ(h ′ ) ≤ ∥h -h ′ ∥ 2 .

We claim that if ϕ(h) = f (h) -D∥∇f (h)∥ and ψ(h ′ ) = f (h ′ ) then the pair Φ, Ψ satisfies ϕ(h) -ψ(h ′ ) ≤ ∥h-h ′ ∥ 2 2 . Indeed,

ϕ(h) -ψ(h ′ ) = f (h) -f (h ′ ) -D∥∇f (h)∥ = f • g(1) -f • g(0) -D∥∇f (h)∥,
where g(t) = th + (1 -t)h ′ . Then, by the fundamental theorem of calculus, we have

ϕ(h) -ψ(h ′ ) = 1 0 (f • g) ′ (t)dt -D∥∇f (h)∥ = 1 0 ∇f th + (1 -t)h ′ , h -h ′ dt -D∥∇f (h)∥.
We now control the last term using that ∥h -h ′ ∥ ≤ D and Cauchy-Schwarz:

ϕ(h) -ψ(h ′ ) ≤ 1 0 ∇f th + (1 -t)h ′ , h -h ′ dt -∇f (h), h -h ′ = 1 0 ∇f th + (1 -t)h ′ -∇f (h), h -h ′ dt.

C.7. Proof of Theorem 13

Proof We start from Theorem 11, using that ∆ Sm is G-gradient-Lipschitz for any m to obtain:

E h∼Q [∆ Sm (h)] ≤ G 2 W 2 2 (Q, P ) + E h∼P [∆ Sm (h)] + DE h∼Q [∥∇∆ Sm (h)∥]
The only thing left to control is E h∼P [∆ Sm (h)]. For this, we use that P alongisde the supermartingale concentration inequality of Chugg et al. (2023, Corollary 17) instantiated with prior equal to posterior, i.i.d. data showing that, for any λ > 0, with probability at least 1 -δ,

E h∼P [∆ Sm (h)] ≤ log(1/δ) λ + λ 2 E h∼P E z∼D [ℓ(h, z) 2 ].
The last term on the right-hand side is bounded by σ 2 by assumption, then, taking λ = 2 log(1/δ) σ 2

gives finally E h∼P E z∼D [ℓ(h, z) 2 ] ≤ 2 log(1/δ)/m, concludes the proof.

Figure 1 :

 1 Figure 1: Evolution of the test risks (with the 01-loss and the bounded cross-entropy loss) and the value of C during the training phase.

E

  z∼D [ℓ(h, z) 2 ] = E z∼D  Var h∼P -γ RSm (ℓ(h, z)) + E h∼P -γ RSm [ℓ(h, z)]

  y).Appendix B. PAC-Bayes bounds for Lipschitz losses through log-Sobolev inequalities Extending Catoni's bound to Lipschitz losses. A well-known relaxation of Catoni (2007, Theorem 1.2.6) (see e.g.Alquier et al., 2016, Theorem 4.1) holding for subgaussian losses has been widely used in practice as a tractable PAC-Bayesian algorithm exhibiting a linear dependency on the KL divergence. We exploit below a consequence of the Herbst argument as stated, e.g., inLedoux (2006, Section 2.3), stating that a L-Lipschitz function of a random variable following a distribution D being L-Sob(c LS ) is L c LS (D) subgaussian. This yields the following corollary.

  KL P-γ RSm , P ≤ c LS (P) E RSm ∥∇ h RSm (h)∥ 2C.4. Proof of Theorem 10Proof We start again fromChugg et al. (2023, Corollary 17) instantiated with a single λ, i.i.d. data and a prior P. Then with probability at least 1 -δ, for any posterior Q and m > 0:

		h∼P [∥∇g(h)∥ 2 ]
	=	γ 2 c LS (P) 4	E h∼P	∥∇ h RSm (h)∥ 2 g 2 (h)
	=	γ 2 c LS (P) 4	E h∼P	∥∇ h RSm (h)∥ 2 dQ dP	(h)
	= h∼P -γ R D (Q) ≤ RSm (Q) + γ 2 c LS (P) 4 E KL(Q, P) + log(1/δ) λm +	λ 2	E

h∼Q E z∼D [ℓ(h, z) 2 ] ,

Acknowledgments

Paul Viallard and Umut S ¸ims ¸ekli are partially supported by the French program "Investissements d'avenir" ANR-19-P3IA-0001 (PRAIRIE 3IA Institute). Umut S ¸ims ¸ekli is also supported by the European Research Council Starting Grant DYNASTY -101039676. Benjamin Guedj acknowledges partial support from the French Research Agency through the programme "France 2030" and PEPR IA on grant SHARP ANR-23-PEIA-0008.

Appendix A. Supplementary background A.1. Additional details on Poincaré and Log-Sobolev inequalities.

Proof of proposition 3. Proof We define P 1 such that dP 1 (h) ∝ exp(-V (h) -γ RSm (h))dh, then note that, by convexity assumption over ℓ 1 , Hess(V +γ RSm ) ⪰ 1 c LS Id. Then, applying Chafaï (2004, Corollary 2.1), we know that P 1 satisfies a Poincaré inequality with constant c LS (P).

Finally, defining P 2 such that dP 2 (h) ∝ exp(-γ m m i=1 ℓ 2 (h, z i )), thanks to the boundedness of ℓ 2 , we use Guionnet and Zegarlinksi (2003, Property 2.6), which ensure that P 2 = P -γ RSm dP 1 (h) satisfies a Log-Sobolev inequality with constant 2c LS (P) exp(4∥ℓ 2 ∥ ∞ )c P (P). Noting that P 2 = P -γ RS concludes the proof.

Proof of Ledoux (2006, Propsition 2.1) We prove here Proposition 14, stated below, showing that Log-Sobolev implies Poincaré.

We then have c

We provide the proof for completeness.

We then apply the Log-Sobolev inequality on 1 + εf :

Note that, by a Taylor expansion, log(1 + εf ) = εf -(εf ) 2 2 + o ε 2 and also that log(1 + ε 2 ) = ε 2 + o(ε 2 ). Then, plugging this into the previous equation gives:

We use that E[f ] = 0 and we then divide by ε 2 . Taking the limit ε → 0 gives:

Using that E[f 2 ] = 1 gives:

Then, for any g ∈ H 1 (Q) applying this proof on f =

concludes the proof.

Then by Cauchy-Schwarz alongside Lipschitz gradient,

We then conclude by applying ( 7) to the pair (2ϕ, 2ψ). The general case with G ̸ = 1 is immediate when considering the pair

C.6. Proof of Corollary 12

Proof We fix R > 0 and we start from Theorem 11 with predictor space H 0 = B(0, R), f being gradient-Lipschitz on this ball and prior and posterior

We first prove that W 2 2 (P R #Q, P R #P) ≤ W 2 2 (Q, P). Let π ∈ Γ(Q, P) being the optimal transport coupling from P to Q, i.e.

Then notice that if we denote by π 1 = (P R , P R )#π, then π 1 ∈ Γ (P R #Q, P R #P) and so:

Using that P R is 1-Lipschitz gives,

Then we need to control W 2 2 (Q, P). To do so, we use the fact that P is L-Sob(c LS ) to affirm, through Otto-Villani's theorem [START_REF] Otto | Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality[END_REF], Theorem 1) that the following holds: W 2 2 (Q, P) ≤ c LS (P) 2 KL(Q, P). This concludes the proof.


Nassima Ait Mansour
email: nassima.aitmansour-etu@etu.univh2c.ma

Hanae Sbai
email: hanae.sbai@fstm.ac.ma

Karim Baina
email: karim.baina@um5.ac.ma

Manal Hilali
email: m.hilali@uhp.ac.ma

Abdellah Ezzati

Said Ben

Alla Hassan

Omaima El Alaoui-Elfels
email: omaima.ee@gmail.com

Taoufiq Gadi
email: gtaoufiq@yahoo.fr

Abdelhak Bassir
email: abdelhak.bassir@uhp.ac.ma

Soufiane Ait Elghazi

Youssef Balouki
email: youssef.balouki@uhp.ac.ma

Amal El Belghiti
email: amal.elbelghiti@gmail.com

Lahcen Belouaddane
email: lahcen.belouaddane@uhp.ac.ma

Mehdi Ait Said

Abderrahim Marzouk
email: abderrahim.marzouk@uhp.ac.ma

Halil Eren
email: halileren7@gmail.com

Cansu Bozkurt
email: cansubozkurt@ardahan.edu.tr

Mahmut Firat
email: mahmut.firat@inonu.edu.tr

Faisal Rehman
email: faisalrehman0003@gmail.com

Muhammad Muneer
email: muneer0206@gmail.com

Muhammad Hamza Sajjad

Intelligent Microservice Discovery using Data Mining Methods

Keywords: Microservice Decomposition, Migration, Process Mining, Monolithic System, Dynamic Analysis Electroencephalogram (EEG), deep learning, feature extraction, neural network, DEAP dataset, CNN, LSTM Cyber security, Deep learning, Malware Visualization, Malware classification, Capsule Network, Residual neural network, Dynamic routing, Imbalanced dataset Recommender Systems, Context-aware, Multi-Criteria, Deep Learning, Deep Neural Network Data Warehouse (DW), ETL, Data Quality (DQ), Data Quality Dimensions. Securing Smart Cities: DevSecOps in Distributed Environments DevOps, DevSecOps, Distributed Architectures, Security Water loss management, Reel losses, Weight coefficient calculation, Multi-criteria decision making, AHP Computational Intelligence, Big Data, Internet of Things, Machine Learning

Due to the advantages, they offer over conventional monolithic systems, microservices have gained popularity in recent years. The challenge for businesses in today's fast-paced digital economy is to develop and manage sophisticated software systems that can efficiently expand, and provide seamless user experiences. Using microservices, businesses can break up large, complex applications into smaller, easier-to-manage components. One of the key problems is how to transition from a monolithic application that already exists to a microservice architecture. Following an analysis of the selected studies to identify distinctive migration methodologies, we can provide the following three migration techniques such as Model-Driven (MD), Static analysis (SA), Dynamic analysis (DA). Software architects frequently do the decomposition manually based on static analysis using programs like Structure 1011. However, runtime dependencies like frequent method calls can have an influence on both maintainability and performance. Static dependency analysis tools cannot capture dynamic system behavior. Using process mining, we may obtain a respectable application decomposition. By utilizing our (DA)-based methodology, the decomposition may be done based on the various user requests, as part of the mining process, this data could be recorded in an event log and used with a scalable business system.

Preface: The 2 nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23)

The Second Edition of the International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) is set to take place on December 19, 2023, at FST Settat, Morocco. This year, the event will be centered around the theme "Predict and Protect: AI and IoT Strategies for Disaster Prevention," focusing on discussing state-of-the-art AI and IoT-based solutions for better prevention against natural disasters.

The main objective of this workshop is to bring together innovative scientists, professors, research scholars, students and industrial experts to exchange their experiences and research results on all aspects of Intelligent Systems and their impact in Disaster Management. It is also an opportunity to discuss and interchange about the real needs of Smart Cities in order to contribute in realizing smart communities of the future. The Workshop covers all topics of Smart Cities, Artificial Intelligence, Internet of Things, Communication Networks, Data Analytics and Disaster Management Systems.

In more details, the conference brought together more than 30 participants. There were three plenary sessions and three invited speakers from different countries sharing their expertise and discussing new topics.

We would like to express our appreciation and gratitude to all invited and keynote speakers, as well as all participants for the highly animated and insightful discussions. We also thank our reviewers for maintaining a good quality standard of the manuscripts that were accepted for participation. We do hope that the proceedings will serve as a valuable reference and be able to stimulate further research in computing intelligence, communication areas and disaster management systems.

In contrast to some studies that adhered to fixed window sizes, our approach acknowledges the impact of varying window sizes on accurate emotion detection. Particularly, we scrutinize the baseline signal (3s), emphasizing the potential distortion in understanding its influence when using extended window lengths. Acknowledging the demonstrated efficacy of considering the first three seconds of the signal, our methodology involves segmenting the signal into pre-trial and trial phases, extracting meaningful information crucial for enhancing model accuracy.

Expanding the dataset and evaluating its impact on prediction accuracy, we employ a ranging window with values [0.25, 0.5, 0.75, 1, 1.5]. The effectiveness of combining four frequency bands -θ, α, β, and γis demonstrated. Employing the Butterworth filter, we decompose each segment into these bands, while δ is not used since it is reserved for deep sleep. Post-processing involves averaging the baseline signal, subtracting it from the trial signal, and extracting frequency characteristics (differential entropy DE, fast Fourier transform FFT, discrete wavelet transform DWT, and power spectral density PSD). This yields feature segments for each frequency band, mapped into 2D vectors and subsequently stacked into a 3D vector.

Transitioning to the CNN architecture, our model extracts frequency and spatial information from each temporal slice. The CNN comprises four convolutional layers, one fully connected layer, and one max-pooling layer. The initial layer houses 64 feature mappings with a (5, 5) filter size. The subsequent layers employ (4, 4) filters, progressing from 128 to 256 feature maps, and concluding with 64 feature mappings and a (1, 1) filter. Recognizing the dynamic content of EEG signals and the limitations of CNNs in capturing additional information, we incorporate an RNN with LSTM cells to extract temporal characteristics.

For temporal dependencies, an LSTM layer with 128 memory cells is employed. Utilizing the extensively studied DEAP dataset, our methodology is validated. This dataset includes EEG signals from 32 individuals who watched 40 one-minute music videos, followed by self-

Enhancing Model Explainability in MLOps using Model-Driven Engineering Paradigms

Mohamed Amine El Youssr, Mahmoud Elhamlaoui, Youness Laghouaouta Mohamed 5 University (UM5), National High School of Computer Science & Systems'Analysis (ENSIAS), Rabat, Morocco (E-mails: ma.elyoussr@gmail.com, mahmoud.elhamlaoui@gmail.com)

Abstract

The rapid integration of machine learning models into real-world applications within Machine Learning Operations (MLOps) has brought about remarkable progress and transformation. However, this progress is accompanied by a pressing challenge: the intricate nature of many advanced models presents a significant hurdle in understanding and trusting their predictions. This challenge underscores the critical need to enhance model explainability within MLOps workflows, ensuring that stakeholders can decode and trust the decisions made by these advanced algorithms. This paper delves into the intricate details of enhancing model explainability in Machine Learning Operations (MLOps) by harnessing specific Model-Driven Engineering (MDE) paradigms. We focus on the application of domain-specific languages (DSLs) tailored for MLOps, intricate model transformations, and the utilization of metamodels to capture the nuances of machine learning models. Our study showcases the practical implementation of these paradigms, demonstrating their efficiency in augmenting interpretability and transparency in MLOps frameworks. Through an in-depth exploration of DSLs, we clarify how these languages can encapsulate domain-specific knowledge, facilitating the creation of more understandable and context-aware models. The paper highlights real-world examples of model transformations, illustrating how MDE techniques can be employed to convert complex, black-box models into more interpretable representations without sacrificing performance. Additionally, we delve into the role of metamodels in capturing the underlying structures of machine learning models, enabling a deeper understanding of model behavior and facilitating enhanced explanations. Furthermore, the study provides insights into the integration of these technical components within existing MLOps workflows, addressing practical considerations and potential challenges. By offering a detailed technical roadmap for incorporating MDE paradigms, this paper aims to empower researchers and practitioners to leverage advanced engineering techniques for achieving superior model interpretability in MLOps environments. Keywords MDE, DSLs, XAI, MLOps.

Reusable Microservices Framework

Mehdi Ait Said Hassan First University, Settat, Morocco (E-mail: lahcen.belouaddane@uhp.ac.ma) 3 Hassan First University, Settat, Morocco (E-mail: abdellah.ezzati@uhp.ac.ma)

Abstract

In recent years, Microservice Architecture (MSA) has gained widespread acclaim for its scalability, flexibility, and modularity, positioning itself as a preferred choice for organizations navigating the complexities of modern software systems [1]. Concurrently, approaches like Model-Driven Development (MDD) and Domain-Driven Design (DDD) have garnered attention for their potential to refine system design and streamline the development process [2]. Our work explores the transformative impact of MSA on software development, emphasizing reusability as a cornerstone principle. Reusable components, ranging from code snippets to entire microservices, serve as foundational elements, encapsulating the collective wisdom of developers and significantly reducing development time and effort. MSA's agile and scalable nature, coupled with its emphasis on loose coupling and high cohesion, fosters a culture of efficiency and consistency within software engineering teams.

To comprehensively address the challenges associated with reusability in MSA, we introduce the Reusable Microservices Framework (RMF), a systematic approach merging principles from MDD and DDD. The RMF, by uniting the centrality of models in MDD and the domain-focused methodologies of DDD, provides a comprehensive solution to enhance the reusability of microservices. Beyond the realm of traditional software development, our work extends its focus to the domain of smart cities, where the importance of efficient and reusable software solutions is paramount. The RMF emerges as a pivotal contribution, addressing critical challenges within MSA and offering a systematic approach to enhance reusability, streamline development processes, and foster collaboration among stakeholders. In the context of smart cities, where diverse systems and services converge, the modular and agile nature of microservices, as promoted by the RMF, aligns seamlessly with the dynamic requirements of urban living. By optimizing development efficiency, reducing time and costs, and promoting interoperability, the RMF sets the stage for the creation of innovative and adaptable smart city solutions that efficiently navigate the complexities of urban management.

Keywords

Microservice Architecture, RMF, DDD, MDD, Reusability.

  mail: m.aitsaid@uhp.ac.ma) 2

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 2

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 4

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 9

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 11

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 15

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP 27

nd International Workshop on Intelligent Systems for Sustainable Smart Cities (I3SC'23) PROCEEDINGS OF I3SC'23 WORKSHOP

We propose a new framework for the automatic discovery of IoT-aware business processes. This framework consists of three elements: model-based transformation, semantic enrichment, and discovery. The first stage converts sensor logs into event logs using a model-driven architecture. The next phase enriches event logs with a structured set of terms and concepts, while discovery creates an intelligent environment aware of IoT-related processes using process mining techniques. Keywords Internet of Things (IoT), Business Process Management (BPM), Process Mining (PM), Process -aware-IoT.

Business Process Management in an IoT Environment via Process Mining

Iman El Kodssi 1 , Hanae Sbai 2 Mathematics, Computer Science and Application Laboratory FST Mohammedia, Hassan II University of Casablanca,Morocco (E-mail: elkodssi.iman@gmail.com) (E-mail: hanae.sbai@fstm.ac.ma)

Abstract

The evolution of information technologies since the end of the 20th century has led companies to adopt these technologies in their management. As business processes became more intelligent, process mining techniques emerged. They have made it possible to automate complex tasks and improve decision-making.

From another point of view, the growth of connected devices in IoT domains has created new opportunities for the advancement of automated business process management.

The main challenge will be to use process mining on sensor logs, which store raw information produced by IoT devices. By converting sensor logs into event logs, numerous methods have been developed in the literature to automatically discover business processes. All these approaches fail to identify the sensors and their data in the final process model.

Abstract

This article focuses on citizen participation in the decision-making process regarding the management of Smart Territory. The purpose is to demonstrate the extent to which the integration of this participatory approach in Smart mobility domain, will enhance the performance of mobility services while mastering the social costs caused by proliferating urbanization.

Territorial management has indeed become a complex governance issue. In the meantime, the progress in information and communication technologies (Artificial Intelligence, Internet of Things, Open Database…) is providing new solutions for a better Territory management, making many complex tasks a snap.

However, territorial management is not only about the smooth running of infrastructure, but also about satisfying subjective needs of local residents that Artificial Intelligence cannot predict. That is why, the new Smart Citizen paradigm has emerged as it advocates putting citizens needs at the heart of territorial management.

Using their smartphones, citizens can play the role of sensors and provide personalized information that cannot be generated by other smart sensors connected to the Internet of Things. The latter is the network of interconnected devices in the Smart Territory ecosystem, enabling them to communicate and exchange data among them and the cloud.

The territory is characterized by several aspects. Our work concentrates on the most synergistic aspect of the territory, that enables the blood to flow through the urban body. Obviously, without mobility, the territory is bled dry.

To demonstrate how active citizen participation in the design of Smart Mobility solutions will enable the latter to better meet expectations in terms of sustainable transport. We will conduct a qualitative study based on analyzing firstly, Smart Citizen concept and its foundations and secondly, Smart Mobility strategies that relay on Intelligent transport system. In order to uncover the opportunities that can be brought about by adopting this concept in transport ecosystem. Benguerir, Morocco (E-mail: b.lakssir@mascir.ma) Abstract Nowadays, as the population is growing at an incredible rate, the adoption of AI and IoT is necessary for ensuring sustainable development in smart cities, which calls for smart agriculture.

Keywords

Weeds are one of the disasters threatening agriculture, which means that weeding is a big issue confronting all farmers. This task involves first removing all undesirable plants that compete with crops for nutrients. These tedious tasks need to be automated or done using intelligent systems that can work and make decisions in real time, rather than respecting the soil and all other requirements to protect the field. In this case, weeding robots based on IoT have already been developed and available around the world since 2011. After more than a decade, the robots are still under modification to improve their performance while preventing the field and guaranteeing sustainable smart cities.

The goal of the workshop is to highlight the state-of-the-art weeding robot, starting with his specific weeding method and finishing with the sensors or components used. In this second session, we will propose as a suggestion a modular architecture for a weeding robot in the design process. The last section is dedicated to showing how AI and IoT can preserve sustainable smart cities in the digital age. Keywords Artificial Intelligence (AI), Internet of Things (IoT), Agriculture, Weeding Robot, Sustainable Development, Smart cities, Smart Agriculture.

Blockchain-Based E-Voting: Ensuring Electoral Process Integrity and Accessibility

Rabia Fatih 1 , Sara Arezki 2 , Taoufiq Gadi

Abstract

The integration of blockchain technology into electronic voting (e-Voting) systems is a revolutionary solution aimed at resolving persistent issues related to fraud, tampering, and the absence of transparency in electoral procedures. This abstract provides an analysis of the transformative potential of blockchain technology, highlighting its capacity to establish an unalterable and transparent ledger of votes. By doing so, it enhances the credibility and verifiability of election results. In addition to its inherent security features, blockchain technology allows for the secure facilitation of remote and tamper-resistant voting, which has the potential to enhance accessibility for a wider range of voters.

Although the potential advantages of implementing blockchain technology in electronic voting systems are significant, there are still challenges that need to be addressed regarding governance, adoption, and technical scalability. Furthermore, it is imperative to prioritize the development of user-friendly interfaces and adhere to regulatory frameworks in order to facilitate widespread acceptance and successful implementation. The purpose of this discussion is to delve into the complex realm of blockchain-based electronic voting (e-Voting). Our focus will be on examining practical applications, innovative approaches, and current research endeavors in this field. This session aims to analyze the relationship between blockchain technology and the electoral process, with the goal of developing a thorough comprehension of its impact on democratic practices. The objective is to facilitate the establishment of voting systems that are more resilient, inclusive, and transparent.

Keywords

Blockchain, E-voting, Decentralization, Consensus.

Tracking and Tracing Containers Model Implemented using Blockchain Technology

Abstract

Blockchain technology is an emergent technology that changes how services and operations work in different fields. It was implemented in different sectors due to its characteristics of transparency and automatic validation with cryptographic algorithms and consensus protocols One of the critical sectors, that needs an intermediary actor in data processing is the portal sector, where most services include the interference of a third party: customs operators, or suppliers, whether in shipping or data asset tracking. This work will focus on tracking and tracing containers in a smart Moroccan port, we present a conception model and Blockchain module for tracking and tracing containers using a consensus algorithm.


Muntadhar Al Moosawi

Aaron Iding

Paul Armstrong

Thita Chiasakul

Robert Campbell

Caterina Casari

Impact of the COVID-19 pandemic on education and clinical training

  L'archive ouverte pluridisciplinaire

The Impact of COVID-19 pandemic on education and clinical training

Since late 2019, the COVID-19 pandemic has forced the world to develop new lifestyles to control the spread of the virus. Education, in general, has been significantly affected by the pandemic. To cope with the changes imposed by the pandemic, universities and education and research institutions had to redesign their curricula and learning strategies to be able to harmonize with the necessary transformations of their education style. [START_REF] Lucey | The transformational effects of COVID-19 on medical education[END_REF] At the height of the first wave of infections, restrictions were greatest, resulting in the partial or complete closure of teaching rooms, libraries, and laboratories. Such changes posed significant challenges to learning methods and environments, and left trainees with reduced core skills compared with their colleagues, who had completed their trainings in pre-COVID years. For example, medical and nursing students received limited clinical exposure because of suspension of trainee participation in hospital rounds, whereas graduate and undergraduate science students were left with limited experience of practical laboratory techniques following cessation of laboratory and research activities. Similarly, a switch from in-person classes to virtual sessions brought technical issues and has proven problematic for audience contribution and two-way interactions. Furthermore, direct and regular mentor/mentee exchanges were interrupted because of redeployment of the institutional workforce to manage the rapid increase in COVID-19 patients and COVID-19 related research.

Although these transformations were and are challenging, necessity has driven their adoption. Indeed, uncertainty over the future course of the pandemic means these challenges are likely to continue for longer time. In response, educational communities are rightly looking for guidance to further improve their teaching methods. To provide such guidance, Almarzooq et al. identified four key features of successful virtual learning: integration, collaboration, education, and communication. [START_REF] Almarzooq | Virtual learning during the COVID-19 pandemic: a disruptive technology in graduate medical education[END_REF] Appropriate implementation of these features is expected to result in more effective and sustainable online learning experiences.

After an initially difficult transition period, we are already benefiting from increased online offerings with a tremendous rise in the number of educational sessions. Many online educational platforms have been developed by commercial and nonprofit organizations to serve as valuable sources for educational material. These alternative learning media offer additional formats and variety to conventional in-person lectures. ISTH webinars, for example, provide a wide range of audience and scholastic content that can be accessed at any time, from anywhere in the world and, importantly, at no additional cost. The advent of national and international hybrid (in-person and virtual) conferences will also make it easier and less costly to attend. Such changes, especially when coupling content provision with social media platforms, further improves educational accessibility and engagement.

However, this substantial increase in opportunities and surge of information can be potentially overwhelming; identifying highquality, valuable content can be challenging. "Webinar fatigue" is a serious problem that can result in physical and psychological trauma. [START_REF] Sharma | Webinar fatigue: fallout of COVID-19[END_REF] A blurred work-life balance also poses disruption of lifestyles and requires stress management. It is important going forward, as a scientific and medical community, that we become conscious of these negative aspects and be sympathetic when designing educational timetables to mitigate against such detrimental effects.

Finally, the economic impact of the COVID-19 pandemic on education and training should not be underestimated. Although the pandemic initially brought about additional funding opportunities, charity incomes and research expenditures are projected to be reduced for the foreseeable future. Unless preventative actions are taken, this could result in decreased numbers of doctoral students and early career researchers, potentially hindering science and clinical care long after this pandemic passes. In conclusion, the COVID-19 pandemic has caused rapid and global transformation within education and clinical trainings. Many changes bring along improvement and are likely to stay. The challenge now is to adapt and evolve these new capabilities alongside pre-COVID teaching methods to create accessible, engaging, interactive, and sustainable educational environments. In doing so, we may turn the challenges of today into the opportunities of tomorrow.

Muntadhar Al Moosawi 1

Aaron Iding 2 Paul Armstrong [START_REF] Sharma | Webinar fatigue: fallout of COVID-19[END_REF] Thita Chiasakul 4 Robert Campbell 5 Caterina Casari 6 For the ISTH Early Career Committee*


Synergistic retrieval of H 2 O vapor in Mars' atmosphere:

the path to a systematic 3D exploration of H 2 O vapor F. Montmessin 1 , S. Ferron 2 1 CNRS, LATMOS, Guyancourt, France , 2 ACRI-ST, Guyancourt, France Summary. Understanding the contemporary water cycle on Mars is a major goal in Mars' climate research. Our study has been conducted with the motivation of optimizing the scientific return of nadir observing instruments which have performed water vapor analysis at various wavelengths on Mars. First, we have evaluated the theoretical performances of jointly analyzing water vapor retrieval in both Thermal Infrared (TIR) and near-Infrared (NIR) ranges (i.e. less than 3 μm). We then proved, by applying our method to actual Mars Express data, that a simultaneous retrieval in the TIR and NIR of the water vapor abundance is susceptible to provide a very strong constraint on the shape of the water vapor profile. Up to now, only a columnintegrated abundance has been derived from nadir observing instruments whereas it appears now possible to derive a second information on water vertical confinement, opening the path to a 3D exploration of water vapor on Mars.

Introduction.

A Synergistic retrieval of trace gases (such as CO or CH4) in the Earth atmosphere has proven very effective for constraining the vertical partitioning of these gases and might be used to identify potential sources at the surface [START_REF] Worden | Observations of near-surface carbon monoxide from space using MOPITT multispectral retrievals[END_REF][START_REF] Worden | Observations of near-surface carbon monoxide from space using MOPITT multispectral retrievals[END_REF]. Guided by this innovation, we have decided to apply a similar approach to the retrieval of water vapor when observed a nadir looking mode on Mars. A number of orbital assets have accomplished the detailed exploration of the nadir column abundance of water vapor on Mars, allowing the continuous monitoring of the water vapor temporal and horizontal distribution for almost 15 consecutive terrestrial years [START_REF] Smith | Interannual variability in TES atmospheric observations of Mars during 1999-2003[END_REF][START_REF] Fedorova | Mars water vapor abundance from SPICAM IR spectrometer: Seasonal and geographic distributions[END_REF][START_REF] Trokhimovskiy | Mars' water vapor mapping by the SPICAM IR spectrometer: Five martian years of observations[END_REF]. This dataset so far has only delivered an information on the vertically integrated amount of water, yet this information was obtained with a variety of instruments operating at different wavelengths. Indeed, it is possible to "see" water vapor through its absorption/emission in the thermal infrared with the Planetary Fourier Spectrometer (PFS) and through its absorption in the NIR domain with three different instruments (SPICAM, OMEGA and PFS). We undertook an examination of the contribution of a NIR / TIR synergy to the restitution of water vapor in the atmosphere of Mars. This approach is an important task in view of cross-comparing the various climate series which have been produced for water vapor to date. Our work was divided in two stages: first, establish the theoretical relevance of this method for currently operating instruments on Mars, and second test and validate the performances predicted at completion of the first stage.

NIR-TIR Synergy methodology.

The work presented below is limited to the cases of the SPICAM and PFS-LW instruments. We relied on an instrumental model that we designed according to the spectral resolutions and mean SNR of the different sensors. Synthetic spectra were produced from atmospheric modeling extracted from a Martian climate database. A direct model has been developed to account for molecular absorption and aerosol scattering. It enabled us to estimate the sensitivities of the radiances to the various geophysical parameters. We have implemented a statistical approach to combine these sensitivities with the noise models of each instrument and to estimate errors and resolutions on the vertical steam and temperature profiles. It emerged from this study that the number of pieces of information on the vertical profile of water vapor is around 1 for NIR bands below 1 for TIR bands taken independently. The synergy of the entire band offers an increase by about 50%. This increase is highly dependent on the aerosol opacity estimated in the TIR. It has also been demonstrated that the sounded portions of altitude are different between the NIR bands, which give access to the layer immediately above the surface, while the TIR band is sensitive to the highest altitudes (> 10 km). On this basis, we have undertaken a statistical characterization of an estimator of the water vapor column and we have evaluated its robustness with respect to the a priori knowledge of aerosols. A preliminary approach allowed us to show that the synergy, in order to be significant, requires to have spectra averages to increase SNR. The synergy is shown to render information from an atmospheric portion that extends from the surface up to more than one height scale (see Figure 1), and also reduces the error on the column-integrated abundance estimate. It requires a prior knowledge of aerosols in order to limit estimation bias.

NIR-TIR Synergy Validation.

The dataset used for testing and validating has been established from spectra acquired during Martian Year (MY) 27 by the SPICAM-IR and PFS-LW spectrometers. It consists of an ensemble of L1 products (calibrated spectra with wavelength assignment) selected specifically for this study. Each product corresponds to an ensemble of collocated spectra satisfying the following criteria: (i) a high quality of the individual measurements; (ii) a good geographical and seasonal coverage of the final dataset encompassing various dust or water ice cloud opacity situations; and (iii) a low error modeling due to surface inhomogeneity (albedo and emissivity). The results obtained with these tests (see Figure 2 for an example) using actual data confirmed the predictions of our theoretical investigation: the number of independently retrieved parameter increase when adding NIR and TIR together, by an amount of more than 20%, reaching >50%. This characteristic offers an optimized reassessment of the experimental data on Mars' atmospheric water collected from the orbit in a nadir-looking mode. Because the latter is the prime mode of observations of the instruments used to explore Mars, the corresponding data volume and its spacetime coverage by far outmatch those of data collected in dedicated viewing geometry like solar occultations or limb staring and thus offers a rich and exciting opportunity to study a yet unexplored "territory" of Martian observations.

Conclusion.

We have defined and implemented a Bayesian approach to estimate water vapor in the Mars atmosphere from NIR and TIR spectra obtained in nadir geometry. An analysis of the measurements information content showed that, on average, the spectral synergy allows the increase of retrieved information on water vapor (which can be represented by the number of independent parameters) by 20 or 50% depending on the reference case (NIR or TIR). We also demonstrated that the NIR/TIR synergy makes it possible to give a more reliable estimate of the total column while unbiasedly estimate the partial column in the first 5 km of the boundary layer, an atmospheric region long remained unexplored although being known for hosting key mechanisms controlling the fate of volatiles on Mars. Using synergy, the scientific return of nadir observations of current and future missions at Mars can be fully optimized.

Figure 1 :

 1 Figure 1: averaging kernels of water vapor in various configurations (from left to right: NIRfrom SPICAM, TIR from PFS TIR ranges, and synergy NIR+TIR).

Figure 2 :

 2 Figure 2: Plot showing the relative vertical partitioning of water as deduced from the NIR+TIR synergetic method. The quantity displayed, expressed in %, is the 5-km partial column ratioed with the total column abundance of water. Higher ratio values indicate stronger confinement of water near the surface.

Scientific Workshop: "From Mars Express to ExoMars" 27-28 February 2018, ESAC Madrid, Spain Workshop organized by ESA-ESAC & IAA-CSIC, with the support of the ESAC Faculty and the EU H2020 project UPWARDS

Acknowledgements. This work recived fundings from the European Union's Horizon 2020 Programme under grant agreement UPWARDS-633127.


Lydia Ouldouali
email: ouldouali@limsi.fr

Nicolas Sabouret
email: nicolas.sabouret@limsi.fr

Charles Rich
email: rich@wpi.edu

I've Got the Power's Value! A Computational Model to Evaluate the Interlocutor's Behaviors in Collaborative Negotiation Socially Interactive Agents Track

Keywords: Dominance, Reasoning about other, Theory of mind

We present in this paper a simulation-oriented theory of mind model for interpreting behaviors of power during a collaborative negotiation. This model relies on a model of negotiation that allows an agent to express behaviors of power through its strategy of negotiation. Based on the simulation theory, we adapted the decision model of the agent to reason about its interlocutor's behavior. A preliminary evaluation in the context of agent-agent interaction shows that the system correctly predicts the interlocutor's power.

INTRODUCTION

Negotiation is a common task in daily life. People negotiate not only in professional situations (e.g. for a salary increase or a promotion) but also in more simple situations such as choosing the holiday destination. In the last decade, a variety of conversational agents have been created that can negotiate with people [START_REF] Gratch | The Misrepresentation Game: How to win at negotiation while seeming like a nice guy[END_REF][START_REF] Klatt | Negotiations in the context of AIDS prevention: an agent-based model using theory of mind[END_REF][START_REF] David V Pynadath | Are you thinking what I'm thinking? An Evaluation of a Simplified Theory of Mind[END_REF].

Research in social psychology demonstrated that the relation of dominance affects the way that the negotiation process is perceived [START_REF] Van Kleef | Power and emotion in negotiation: Power moderates the interpersonal effects of anger and happiness on concession making[END_REF]. Negotiators build different negotiation strategies depending on their relative dominance which directly influences the outcomes of the negotiation. More More precisely, Tidens [START_REF] Larissa | Power moves: complementarity in dominant and submissive nonverbal behavior[END_REF] showed that dominance complementarity (i.e. one negotiator exhibits dominant behaviors while the other one responds with submissive ones) leads the negotiators to reach mutually beneficial outcomes and increases their reciprocal likings.

In this poster, we present an agent that simulates such a relation of dominance in collaborative negotiation, based on the computational model proposed by [START_REF] Ouldouali | A computational model of power in collaborative negotiation dialogues[END_REF]. We show how this model can be adapted to reason about the interlocutor's power, following a Theory of Mind approach to simulate the mental activity of the interlocutor. We show that the agent can make good predictions in the context of an agent-agent interaction.

NEGOTIATOR AGENT WITH TOM

Our negotiator agent can handle a negotiation to choose one option over several ones, characterized different criteria values. For example, the negotiation for a restaurant considers the criteria Cuisine, atmosphere, price, and location. The agent has a set of binary preferences ≺ i on each criterion i. It communicates with its interlocutor via abstract utterances (i.e. formal speech acts) to either share its preferences or make negotiations moves. An example of a dialogue is presented in Figure 2.

As presented on Figure 1, the decision model takes into account the agent's preferences in addition to a value of power pow ∈ [0, 1]. The negotiation strategy selects an utterance that the agent communicates to its interlocutor.

Based on the simulation theory, we use the agent's decision model to reason about the interlocutor's behaviors during the negotiation: the agent infers its interlocutor's value of power based on the utterances it received during the negotiation. For this purpose, the agent makes assumptions about the interlocutor's preferences and power. The algorithm is as follows:

(1) Build a set H pow of hypothesis about power: h ∈ H pow represents the hypothesis pow = h. In our work, we consider only 9 values:

H pow = {0.
This model, however, requires building an important number of hypotheses for the agent to consider at each turn. Indeed, given a topic with n criteria, assuming that each criterion has k values, the number of hypotheses on the preference set that the agent has to To overcome this limitation, we consider only a partial representation of the interlocutor's preferences. This solution is supported by research in cognitive psychology: Harbers [START_REF] Harbers | Modeling agents with a theory of mind[END_REF] suggests that, in order to simulate another's mental processes, it is not necessary to categorize all the beliefs and desires attributed to that person as such. In other words, it is not necessary to have a complete model of the interlocutor.

Instead of computing all the possible relation of preferences ≺ i , we only compute the set of satisfiable values (i.e. the values that the interlocutor likes). The set of hypotheses to consider is drastically reduced, but 1) we had to adapt the agent's decision model in the simulation Theory of Mind in order to handle the uncertainty of preferences in the reasoning process and 2) the inferred power value might not be correct since it relies on a different model (with uncertainties).

We conducted an experimental study to assess the validity of the decision model with partial preferences. We implemented two agents with this model. At each dialogue turn, each agent will try to predict the behavior of power expressed by its interlocutor. For each agent, we compared the agent's prediction with the effective A: "Let's go to a restaurant at Montparnasse. " B: "Okay, let's go to a restaurant at Montparnasse. " . . . A: "Do you like expensive restaurants?" B: "I don't like expensive restaurants. " A: "Do you like affordable restaurants?" B: "Let's go to the Maison blanche restaurant. It's a modern, cheap French restaurant on the Montparnasse" A: "Okay, let's go to the Maison blanche restaurant. " value of power assigned to the other agent. We further analyzed the speed of the prediction and the timeliness of the algorithm. The results obtained confirmed the accuracy of our model to predict the behavior of power expressed by an interlocutor during a collaborative negotiation. As a perspective, we intend to evaluate our model of decision to be able to simulate a complementary relation of dominance between an artificial agent and a human user. The goal is to study the impact of a complementary relation on the outcome of the negotiation as well as the experience of the negotiation.

Figure 1 :

 1 Figure 1: Model of collaborative negotiation with a model of other

Figure 2 :

 2 Figure 2: Example of dialogue.

  1, 0.2, . . . , 0.9}. (2) For each hypothesis h, build the set of all possible preferences Prec h : the elements p ∈ Prec h are partial orders on the criteria. (3) After each utterance u, remove all elements in Prec h that are not compatible with u. Concretely, if the applicability condition of u is not satisfied in p ∈ Prec h , then p must be removed from the candidate mental states. (4) For each h, generate the corresponding utterance using h as input for the decisional model. (5) Compute a score score(h) based on the size of remaining hypothesis |Prec h | that generate an output similar to the utterance, U tterance ot her stated by the interlocutor. (6) The hypothesis with the highest score is the most probable value for the interlocutor's power value.

	pow ot her = arg max

h

(score(h))