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Computer Science > Machine Learning

arXiv:2109.03839 (cs)
[Submitted on 8 Sep 2021 (v1), last revised 21 Feb 2022 (this version, v3)]

Title:Sqrt(d) Dimension Dependence of Langevin Monte Carlo

Authors:Ruilin Li, Hongyuan Zha, Molei Tao
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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.
Comments: v1 submitted on May 28, 2021 (NeurIPS 2021 deadline); v2 added an important reference and discussions; v3 is the camera ready version
Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR); Statistics Theory (math.ST); Machine Learning (stat.ML)
Cite as: arXiv:2109.03839 [cs.LG]
  (or arXiv:2109.03839v3 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2109.03839
Journal reference: ICLR 2022

Submission history

From: Molei Tao [view email]
[v1] Wed, 8 Sep 2021 18:00:05 UTC (499 KB)
[v2] Thu, 23 Sep 2021 17:59:14 UTC (503 KB)
[v3] Mon, 21 Feb 2022 01:26:40 UTC (543 KB)
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