Séance du 17 février 2020

Séance organisée par Judith Rousseau et Umut Şimşekli.

Lieu : IHP, amphi Hermite

14.00 : Kweku Abraham (Université Paris-Saclay)

Titre : The Bayesian approach to statistical inverse problems

Résumé : Given a "forward map" which has good continuity properties, but whose inverse is badly behaved, a statistical inverse problem seeks to recover an input from a noisy observation of the corresponding output. I will explain why a Bayesian approach to such problems is natural, and sketch how four components -- estimation at the "regression" level, continuity of the forward map, "stability" estimates, and a priori smoothness assumptions -- combine to yield consistency of this approach. I will outline the statistical Calderon problem as an example of an inverse problem, and explain how these four elements can be achieved there.

15.00 : Lénaïc Chizat (CNRS et Université Paris-Saclay)

Titre : Analysis of gradient methods for the optimization of wide two layer neural networks

Résumé : Gradient-based optimization algorithms applied to artificial neural networks with many parameters typically lead to models with good train and test performance. While some statistical results suggest that two-layer neural networks have the potential to outperform simpler models such as linear regression and its "kernel" extensions, it is not clear whether these results apply to models as used in practice because of issues such as non-convexity or lack of regularization. In this talk, I present a line of work that partially bridges this gap by analyzing the dynamics of training when the width of the hidden layer goes to infinity. We show that this mean-field dynamics follows a "Wasserstein gradient flow" which, in some cases, converges globally and leads to a specific "implicit bias". Along the way, we discuss the statistical implications of these optimization results. This talk is based on joint work with Francis Bach and Edouard Oyallon

16.00 : Anne Sabourim (Télécom Paris)

Titre : Principal Component Analysis for Multivariate Extremes

Résumé : It is well known that the first order behavior of multivariate regularly varying random vectors above large radial thresholds is determined by the homogeneous limit measure. If, for a high dimensional vector, the support of this measure is concentrated on a lower dimensional subspace, identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we consider applying standard Principal Component Analysis (PCA) to a rescaled version of radial thresholded observations. Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radialthresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are re-scaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection subspace. Numerical experiments illustrate the relevance of the proposed framework for practical purposes. This work is joint work with Holger Drees.