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

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.