Couplings of the Random-Walk Metropolis algorithm

TL;DR

This paper investigates various coupling strategies for the Random-Walk Metropolis algorithm, analyzing their theoretical properties and impact on chain meeting times to improve MCMC convergence efficiency.

Contribution

It introduces new coupling designs based on geometric, optimal transport, and maximality principles, providing insights into their effectiveness for MCMC convergence.

Findings

Couplings based on optimal transport can reduce meeting times.

Maximal couplings improve convergence speed.

General principles for designing effective couplings are proposed.

Abstract

Couplings play a central role in contemporary Markov chain Monte Carlo methods and in the analysis of their convergence to stationarity. In most cases, a coupling must induce relatively fast meeting between chains to ensure good performance. In this paper we fix attention on the random walk Metropolis algorithm and examine a range of coupling design choices. We introduce proposal and acceptance step couplings based on geometric, optimal transport, and maximality considerations. We consider the theoretical properties of these choices and examine their implication for the meeting time of the chains. We conclude by extracting a few general principles and hypotheses on the design of effective couplings.