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 et.al, 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.

翻译：一般状态空间马尔可夫链的收敛速率分析在马尔可夫链蒙特卡洛和算法分析（用于计算显式收敛界）等领域具有根本重要性。然而，该问题 notoriously 困难，因为传统的分析方法通常无法为现实中的马尔可夫链生成具有实际用途的收敛界。我们提出了深度收缩漂移计算器（DCDC），这是首个通用的、基于样本的算法，用于在 Wasserstein 距离下界定马尔可夫链向平稳分布的收敛性。DCDC 包含两个组成部分。首先，受（Qu 等人，2023）中新收敛分析框架的启发，我们引入了收缩漂移方程（CDE），其解可导出一个显式的收敛界。其次，我们开发了一种高效的基于神经网络的 CDE 求解器。借助这两个组件，DCDC 求解 CDE 并将其解转换为收敛界。我们分析了该算法的样本复杂度，并通过为源自随机处理网络以及恒定步长随机优化的现实马尔可夫链生成收敛界，进一步证明了 DCDC 的有效性。