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

TL;DR

An explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential $U$ and super-linear gradient of $U$ is proposed and the non-asymptotic analysis of its sampling error in total variation distance is investigated and the smallest number of iterations of the PLMC algorithm is proved to be of order.

Abstract

It is of significant interest in many applications to sample from a high-dimensional target distribution $π$ with the density $π(\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 $\mathcal{O}(d^{\max\{3γ/2 , 2γ-1 \}} h |\ln h|)$, where $d$ is the dimension of the target distribution and $γ\geq 1$ 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 $ε$, the smallest number of iterations of the PLMC algorithm is proved to be of order ${\mathcal{O}}\big(\tfrac{d^{\max\{3γ/2 , 2γ-1 \}}}ε \ \cdot \ln (\tfrac{d}ε) \cdot \ln (\tfrac{1}ε) \big)$. 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 ${\mathcal{O}}\big(\tfrac{d^{3/2}}ε \cdot \ln (\tfrac{1}ε) \big)$. Numerical experiments are provided to confirm the theoretical findings.

Figures (4)

• Figure 1: Probability density of the first component of the double-well model.
• Figure 2: Weak convergence rates of PLMC algorithm of the double well model for $d=6$ (Top) and $d=10$ (Bottom).
• Figure 3: Weak convergence rates of PLMC algorithm of the double well model for $d=50$ (Top) and $d=100$ (Bottom).
• Figure 4: Dimension dependence of weak errors of PLMC algorithm.

Theorems & Definitions (35)

• Remark 2.3
• Example 2.5: Double-well potential
• Theorem 2.6
• Theorem 2.7
• Remark 2.8
• Proposition 2.9
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['lemma:Existence-and-uniqueness-of-the-invariant-measure-paper3']}
• Lemma 3.3