An optimization perspective on log-concave sampling and beyond

Abstract

The primary contribution of this thesis is to advance the theory of complexity for sampling from a continuous probability density over R^d. Some highlights include: a new analysis of the proximal sampler, taking inspiration from the proximal point algorithm in optimization; an improved and sharp analysis of the Metropolis-adjusted Langevin algorithm, yielding new state-of-the-art guarantees for high-accuracy log-concave sampling; the first lower bounds for the complexity of log-concave sampling; an analysis of mirror Langevin Monte Carlo for constrained sampling; and the development of a theory of approximate first-order stationarity in non-log-concave sampling.

We further illustrate the main tools in this work—diffusions and Wasserstein gradient flows—through applications to functional inequalities, the entropic barrier, Wasserstein barycenters, variational inference, and diffusion models.