Résumé du cours

Sampling is a fundamental question in statistics and machine learning, most notably in Bayesian methods. Sampling and optimization present many similarities, some obvious, others more mysterious. In particular, the seminar work of Jordan, Kinderlehrer and Otto (’98) has unveiled a beautiful connection between the Brownian motion and the heat equation on the one hand, and optimal transport on the other. They showed that certain stochastic processes may be viewed as gradient descent over the Wasserstein space of probability distributions. This connection opens the perspective of a novel approach to sampling that leverages the rich toolbox of optimization to derive and analyze sampling algorithms. The goal of this course is to bring together the many ingredients that make this perspective possible starting from the basics and building to some of the most recent advances in sampling.

Sujets abordés

• Convex optimization: algorithms and analysis
• Markov semigroup theory
• Optimal transport
• Wasserstein gradient flows
• Particle methods

Prérequis

Connaissances de base en analyse et en probabilités. Les sujets seront traités en partant du niveau le plus élémentaire.

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