Wasserstein-based methods for convergence complexity analysis of MCMC with applications

TL;DR

This paper introduces Wasserstein distance-based techniques for analyzing the convergence complexity of MCMC algorithms, offering more robust bounds in high-dimensional settings compared to traditional drift and minorization methods.

Contribution

It proposes Wasserstein-based methods that avoid minorization, providing improved convergence bounds for high-dimensional MCMC in Bayesian models.

Findings

Wasserstein bounds are more robust to increasing dimension.

New complexity results for Bayesian probit regression.

Enhanced understanding of MCMC convergence in large-scale problems.

Abstract

Over the last 25 years, techniques based on drift and minorization (d&m) have been mainstays in the convergence analysis of MCMC algorithms. However, results presented herein suggest that d&m may be less useful in the emerging area of convergence complexity analysis, which is the study of how the convergence behavior of Monte Carlo Markov chains scale with sample size, $n$, and/or number of covariates, $p$. The problem appears to be that minorization can become a serious liability as dimension increases. Alternative methods for constructing convergence rate bounds (with respect to total variation distance) that do not require minorization are investigated. Based on Wasserstein distances and random mappings, these methods can produce bounds that are substantially more robust to increasing dimension than those based on d&m. The Wasserstein-based bounds are used to develop strong convergence complexity results for MCMC algorithms used in Bayesian probit regression and random effects models in the challenging asymptotic regime where $n$ and $p$ are both large.