Sqrt(d) Dimension Dependence of Langevin Monte Carlo
Ruilin Li, Hongyuan Zha, Molei Tao
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.
BibTeX
@inproceedings{
li2022sqrtd,
title={Sqrt(d) Dimension Dependence of Langevin Monte Carlo},
author={Ruilin Li and Hongyuan Zha and Molei Tao},
booktitle={International Conference on Learning Representations},
year={2022},
url={https://openreview.net/forum?id=5-2mX9_U5i}
}