From Optimization to Sampling Through Gradient Flows

This article overviews how gradient flows, and discretizations thereof, are useful to design and analyze optimization and sampling algorithms. The interplay between optimization, sampling, and gradient flows is an active research area; our goal is to provide an accessible and lively introduction to some core ideas, emphasizing that gradient flows uncover the conceptual unity behind many optimization and sampling algorithms, and that they give a rich mathematical framework for their rigorous analysis.