On the geometric convergence for MALA under verifiable conditions

TL;DR

This paper establishes verifiable conditions under which the Metropolis Adjusted Langevin Algorithm (MALA) converges geometrically, focusing on tail and smoothness assumptions that are practical and easy to verify.

Contribution

The authors prove V-uniform geometric convergence of MALA under mild, tail, and smoothness conditions on the target distribution, with explicit dependence on step size.

Findings

Geometric convergence guaranteed under verifiable conditions

Explicit bounds depending on the Euler-Maruyama step size

Conditions are common and easy to verify in practice

Abstract

While the Metropolis Adjusted Langevin Algorithm (MALA) is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors' knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish $V$-uniformly geometric convergence for MALA under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the MCMC literature and are easy to verify in practice. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler-Maruyama discretization, which corresponds to the proposal Markov kernel of MALA.