14:00 – 15:00 Po-Ling Loh

Title: Robust empirical risk minimization via Newton’s method

Abstract: We study a variant of Newton’s method for empirical risk minimization, where at each iteration of the optimization algorithm, we replace the gradient and Hessian of the objective function by robust estimators taken from existing literature on robust mean estimation for multivariate data. After proving a general theorem about the convergence of successive iterates to a small ball around the population-level minimizer, we study consequences of our theory in generalized linear models, when data are generated from Huber’s epsilon-contamination model and/or heavy-tailed distributions. We also propose an algorithm for obtaining robust Newton directions based on the conjugate gradient method, which may be more appropriate for high-dimensional settings, and provide conjectures about the convergence of the resulting algorithm. Compared to the robust gradient descent algorithm proposed by Prasad et al. (2020), our algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-order algorithms for convex problems, i.e., quadratic convergence in a neighborhood of the optimum, with a stepsize which may be chosen adaptively via backtracking linesearch.

This is joint work with Eirini Ioannou (Edinburgh) and Muni Sreenivas Pydi (Paris Dauphine – PSL).

15:30 – 16:30 Marta Catalano

Title: Merging rate of opinions via optimal transport on random measures

Abstract:  The Bayesian approach to inference is based on a coherent probabilistic framework that naturally leads to principled uncertainty quantification and prediction. Via conditional (or posterior) distributions, Bayesian nonparametric models make inference on parameters belonging to infinite-dimensional spaces, such as the space of probability distributions. The development of Bayesian nonparametrics has been triggered by the Dirichlet process, a nonparametric prior that allows one to learn the law of the observations through closed-form expressions. Still, its learning mechanism is often too simplistic and many generalizations have been proposed to increase its flexibility, a popular one being the class of normalized completely random measures. Here we investigate a simple yet fundamental matter: will a different prior actually guarantee a different learning outcome? To this end, we develop a new distance between completely random measures based on optimal transport, which provides an original framework for quantifying the similarity between posterior distributions (or merging of opinions). Our findings provide neat and interpretable insights on the impact of popular Bayesian nonparametric priors, avoiding the usual restrictive assumptions on the data-generating process. This is joint work with Hugo Lavenant.

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