Non-asymptotic Analysis of Poisson randomized midpoint Langevin Monte Carlo

TL;DR

This paper analyzes the Poisson Randomized Midpoint Langevin Monte Carlo algorithm, establishing its convergence properties and error bounds for sampling from high-dimensional distributions, and introduces a decreasing-step size variant with near-optimal convergence.

Contribution

It provides the first rigorous analysis of PRLMC's convergence to its stationary distribution and to the target distribution, including error bounds and a nearly optimal decreasing-step size version.

Findings

PRLMC admits a unique stationary distribution under mild conditions.

The convergence rate of PRLMC to its stationary distribution is established.

A decreasing-step size PRLMC achieves near-optimal convergence to the target distribution.

Abstract

The task of sampling from a high-dimensional distribution $\pi$ on $\R^d$ is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on $\R^d$ \begin{align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{2}dB_t, \end{align*} under mild conditions, it admits $\pi(\dif x)\propto \exp(-U(x))\dif x$ as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution $\pi$. In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution $\pi_\eta$ ($\eta$ is the step size) and obtain the convergence rate of PRLMC to $\pi_\eta$ in total variation distance. Then we establish a sharp error bound between $\pi_\eta$ and $\pi$ under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to $\pi$ which is nearly optimal.