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

Abstract

It is of significant interest in many applications to sample from a high-dimensional target distribution π with the density $\pi \left(\text{d}x\right)\propto e^{-U\left(x\right)}\left(\text{d}x\right)$, 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 $O\left(d^{\max \{3\gamma /2,2\gamma -1\}}h\right|\ln h\left|\right)$, where d is the dimension of the target distribution and $\gamma \ge 1$ characterizes the growth of the gradient of U. In addition, if the gradient of U is globally Lipschitz continuous, an improved convergence order of $O\left(d^{3/2}h\right)$ 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 $O(\frac{d^{\max \{3\gamma /2,2\gamma -1\}}}{ϵ}\cdot \ln (\frac{d}{ϵ})\cdot \ln (\frac{1}{ϵ}\left)\right)$. 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 $O(\frac{d^{3/2}}{ϵ}\cdot \ln (\frac{1}{ϵ}\left)\right)$. Numerical experiments are provided to confirm the theoretical findings.