Convergence rate of randomized midpoint Langevin Monte Carlo

TL;DR

This paper analyzes the convergence properties of the randomized midpoint Langevin Monte Carlo (RLMC) algorithm, establishing exponential ergodicity with constant step-size and providing convergence rates for a decreasing-step size variant.

Contribution

It proves exponential ergodicity for RLMC with constant step-size and introduces a decreasing-step size version with explicit convergence rate analysis.

Findings

RLMC is exponentially ergodic with constant step-size.

Decreasing-step size RLMC has quantifiable convergence rates.

The results enhance understanding of RLMC's efficiency in high-dimensional sampling.

Abstract

The randomized midpoint Langevin Monte Carlo (RLMC), introduced by Shen and Lee (2019), is a variant of classical Unadjusted Langevin Algorithm. It was shown in the literature that the RLMC is an efficient algorithm for approximating high-dimensional probability distribution $\pi$. In this paper, we establish the exponential ergodicity of RLMC with constant step-size. Moreover, we design a dereasing-step size RLMC and provide its convergence rate in terms of a functional class distance.