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 $\pi$ with the density $\pi(\text{d} x) \propto e^{-U(x)} (\text{d} x) $, 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 $\mathcal{O}(d^{\max\{3\gamma/2 , 2\gamma-1 \}} h |\ln h|)$, where $d$ is the dimension of the target distribution and $\gamma \geq 1$ characterizes the growth of the gradient of $U$. In addition, if the gradient of $U$ is globally Lipschitz continuous, an improved convergence order of $\mathcal{O}(d^{3/2} h)$ 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 $\epsilon$, the smallest number of iterations of the PLMC algorithm is proved to be of order ${\mathcal{O}}\big(\tfrac{d^{\max\{3\gamma/2 , 2\gamma-1 \}}}{\epsilon} \ \cdot \ln (\tfrac{d}{\epsilon}) \cdot \ln (\tfrac{1}{\epsilon}) \big)$. 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 ${\mathcal{O}}\big(\tfrac{d^{3/2}}{\epsilon} \cdot \ln (\tfrac{1}{\epsilon}) \big)$. Numerical experiments are provided to confirm the theoretical findings.