Projected Langevin Monte Carlo algorithms in non-convex and super-linear setting

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Pang, Chenxu and Wang, Xiaojie and Wu, Yue (2025) Projected Langevin Monte Carlo algorithms in non-convex and super-linear setting. Journal of Computational Physics, 526. 113754. ISSN 0021-9991 (https://doi.org/10.1016/j.jcp.2025.113754)

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Abstract

It is of significant interest in many applications to sample from a high-dimensional target distribution π with the density π(dx)∝e −U(x)(dx), based on the temporal discretization of the Langevin stochastic differential equations (SDEs). In this paper, we propose an explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential U and super-linear gradient of U and investigate the non-asymptotic analysis of its sampling error in total variation distance. Equipped with time-independent regularity estimates for the associated Kolmogorov equation, we derive the non-asymptotic bounds on the total variation distance between the target distribution of the Langevin SDEs and the law induced by the PLMC scheme with order O(d max⁡{3γ/2,2γ−1}h|ln⁡h|), where d is the dimension of the target distribution and γ≥1 characterizes the growth of the gradient of U. In addition, if the gradient of U is globally Lipschitz continuous, an improved convergence order of O(d 3/2h) for the classical Langevin Monte Carlo (LMC) scheme is derived with a refinement of the proof based on Malliavin calculus techniques. To achieve a given precision ϵ, the smallest number of iterations of the PLMC algorithm is proved to be of order [Formula presented]. In particular, the classical Langevin Monte Carlo (LMC) scheme with the non-convex potential U and the globally Lipschitz gradient of U can be guaranteed by order [Formula presented]. Numerical experiments are provided to confirm the theoretical findings.

ORCID iDs

Pang, Chenxu, Wang, Xiaojie and Wu, Yue ORCID logoORCID: https://orcid.org/0000-0002-6281-2229;
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