Is There an Analog of Nesterov Acceleration for MCMC?

TL;DR

This paper explores the connection between Nesterov acceleration and MCMC, showing that an underdamped Langevin algorithm achieves accelerated convergence by formulating sampling as an optimization problem on probability measures.

Contribution

It introduces a novel perspective by framing gradient-based MCMC as optimization on probability spaces and demonstrates accelerated convergence rates for certain nonconvex functions.

Findings

Underdamped Langevin performs accelerated gradient descent in the space of probability measures.

A Lyapunov functional is constructed to analyze convergence.

Accelerated rates are achieved for nonconvex functions using Langevin dynamics.

Abstract

We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback-Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.