Computable Bounds on Convergence of Markov Chains in Wasserstein Distance via Contractive Drift

TL;DR

This paper presents a unified, computable framework for estimating Markov chain convergence in Wasserstein distance, applicable to a wide range of chains including non-contractive ones, using contractive drift conditions and novel techniques.

Contribution

The paper introduces a new framework that provides computable convergence bounds for Markov chains in Wasserstein distance, applicable even to non-contractive chains, with techniques like large M and boundary removal.

Findings

Framework applies to non-contractive Markov chains.

Provides convergence bounds with polynomial to exponential rates.

Enhanced by deep learning techniques in related work.

Abstract

We introduce a unified framework to estimate the convergence of Markov chains to equilibrium in Wasserstein distance. The framework can provide convergence bounds with rates ranging from polynomial to exponential, all derived from a contractive drift condition that integrates not only contraction and drift but also coupling and metric design. The resulting bounds are computable, as they contain simple constants, one-step transition expectations, but no equilibrium-related quantities. We introduce the large M technique and the boundary removal technique to enhance the applicability of the framework, which is further enhanced by deep learning in Qu, Blanchet and Glynn (2024). We apply the framework to non-contractive or even expansive Markov chains arising from queueing theory, stochastic optimization, and Markov chain Monte Carlo.