Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling

TL;DR

This paper provides the first non-asymptotic theoretical analysis of annealed Langevin Monte Carlo, demonstrating its provable benefits for sampling from complex, non-log-concave, multimodal distributions.

Contribution

It establishes a novel oracle complexity bound for annealed Langevin Monte Carlo, offering theoretical guarantees for its efficiency in challenging sampling scenarios.

Findings

First non-asymptotic analysis of annealed MCMC

Oracle complexity bound of O(deta^2A^2/\u03b5^6) for b5^2 accuracy

Demonstrates provable benefits for non-log-concave sampling

Abstract

We consider the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\cal A}^2}{\varepsilon^6}\right)$ for the simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $\pi\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $\beta$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $\pi$ and a readily sampleable distribution.