Deep Learning for Computing Convergence Rates of Markov Chains

TL;DR

This paper introduces DCDC, a novel neural network-based algorithm that provides practical convergence bounds for Markov chains in Wasserstein distance, addressing a key challenge in stochastic process analysis.

Contribution

The paper presents the first general-purpose, sample-based algorithm for bounding Markov chain convergence using a neural network to solve the Contractive Drift Equation.

Findings

DCDC effectively generates convergence bounds for complex Markov chains.

The algorithm demonstrates practical utility in stochastic networks and optimization.

Sample complexity analysis supports the method's efficiency.

Abstract

Convergence rate analysis for general state-space Markov chains is fundamentally important in areas such as Markov chain Monte Carlo and algorithmic analysis (for computing explicit convergence bounds). This problem, however, is notoriously difficult because traditional analytical methods often do not generate practically useful convergence bounds for realistic Markov chains. We propose the Deep Contractive Drift Calculator (DCDC), the first general-purpose sample-based algorithm for bounding the convergence of Markov chains to stationarity in Wasserstein distance. The DCDC has two components. First, inspired by the new convergence analysis framework in Qu, Blanchet and Glynn (2023), we introduce the Contractive Drift Equation (CDE), the solution of which leads to an explicit convergence bound. Second, we develop an efficient neural-network-based CDE solver. Equipped with these two components, DCDC solves the CDE and converts the solution into a convergence bound. We analyze the sample complexity of the algorithm and further demonstrate the effectiveness of the DCDC by generating convergence bounds for realistic Markov chains arising from stochastic processing networks as well as constant step-size stochastic optimization.