# Deep Learning for Computing Convergence Rates of Markov Chains

arXiv: https://arxiv.org/abs/2405.20435 · HTML: https://mlsanity.com/paper/2405.20435/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 30 May 2024.

## Builds on (most cited first, 5 of 5)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Safe Control with Learned Certificates: A Survey of Neural Lyapunov, Barrier, and Contraction methods](https://mlsanity.com/paper/2202.11762/) (2022)
- [Biased Stochastic First-Order Methods for Conditional Stochastic Optimization and Applications in Meta Learning](https://mlsanity.com/paper/2002.10790/) (2020)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Sample Complexity of Sample Average Approximation for Conditional Stochastic Optimization](https://mlsanity.com/paper/1905.11957/) (2019)

## Built on by (most cited first, 4 of 4)

- [Deep Learning for Markov Chains: Lyapunov Functions, Poisson's Equation, and Stationary Distributions](https://mlsanity.com/paper/2508.16737/) (2025)
- [A Variance-Based Analysis of Sample Complexity for Grid Coverage](https://mlsanity.com/paper/2511.17784/) (2025)
- [Deep Learning Method for Stationary Distribution of Reflected Brownian Motion](https://mlsanity.com/paper/2607.08091/) (2026)
- [Machine learning kinetics from molecular dynamics data](https://mlsanity.com/paper/2609.17736/) (2026)

## Abstract (arXiv)

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.