From Optimization to Sampling Through Gradient Flows

TL;DR

This paper explores how gradient flows serve as a unifying mathematical framework for designing and analyzing optimization and sampling algorithms, highlighting their conceptual and analytical advantages.

Contribution

It provides an accessible overview of the role of gradient flows in connecting optimization and sampling, emphasizing their theoretical significance.

Findings

Gradient flows unify optimization and sampling algorithms.

Discretizations of gradient flows are useful for algorithm design.

Gradient flows offer a rigorous framework for analysis.

Abstract

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.