Sqrt(d) Dimension Dependence of Langevin Monte Carlo

TL;DR

This paper provides a refined non-asymptotic analysis of Langevin Monte Carlo, establishing an optimal mixing time bound of O( ext{d}/psilon) in high dimensions, improving previous results and validated by experiments.

Contribution

It introduces a new analytical framework that refines mean-square analysis, leading to an improved mixing time bound for LMC under common conditions.

Findings

Established an O( ext{d}/psilon) mixing time bound for LMC.

Improved previous O(d/psilon) bound, showing optimality in dimension and accuracy.

Validated theoretical results with numerical experiments.

Abstract

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an $\tilde{O}(\sqrt{d}/\epsilon)$ mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known $\tilde{O}(d/\epsilon)$ result and is optimal (in terms of order) in both dimension $d$ and accuracy tolerance $\epsilon$ for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.