Multi-Marginal Couplings for Metropolis-Hastings

TL;DR

This paper develops a novel coupling-based convergence diagnostic for Markov chain Monte Carlo, introducing multi-marginal coupling techniques that improve coalescence rates and reduce meeting times in high-dimensional problems.

Contribution

It introduces a new multi-marginal coupling approach for MCMC convergence diagnostics, with an adaptive Poisson Monte Carlo construction that enhances efficiency in high dimensions.

Findings

Reduces chain meeting times by up to 50% in experiments.

Improves coalescence rates across different dimensions.

Develops an adaptive rule for point process updates.

Abstract

Convergence diagnosis for Markov chain Monte Carlo is a matter of fundamental importance in computational statistics: it determines the resources allocated to a particular sampling problem and influences the practitioner's view of the quality of estimates obtained from a Markov chain. Motivated by this, we contribute to the emerging class of coupling-based convergence diagnostic algorithms. Concretely, we study coupling multiple Metropolis-Hastings chains using multi-marginal coupling. We introduce a natural objective for this setting and establish lower and upper bounds by drawing connections to list-level distribution coupling and distributed pairwise-matching problems. This analysis ultimately leads to a shared-randomness Poisson Monte Carlo construction for coupling multiple Markov chains. In this process, we avoid a key dimension-dependent bottleneck in the runtime complexity of classical Poisson Monte Carlo by developing an adaptive rule for updating the point process, yielding significant gains in high-dimensional settings. Experiments on grand couplings of Markov chains show that our methods improve coalescence rates across dimensions, reducing meeting times by up to 50% compared with existing baselines.