## Sqrt(d) Dimension Dependence of Langevin Monte Carlo

29 Sept 2021, 00:34 (modified: 10 Mar 2022, 23:57)ICLR 2022 PosterReaders: Everyone
Keywords: unadjusted Langevin algorithm / Langevin Monte Carlo, non-asymptotic sampling error in Wasserstein-2 distance, optimal dimension dependence, mean square analysis
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.
One-sentence Summary: The known dimension dependence of LMC is improved, under regularity assumptions, from d to sqrt(d), based on a refined mean square analysis framework.
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