Uncertainty quantification for Markov chains with application to temporal difference learning

2025-02-19

AI Summary

This paper addresses the statistical inference challenge for Markov chain-driven reinforcement learning (e.g., temporal difference learning) under data dependence. It establishes, for the first time, concentration inequalities and Berry–Esseen-type bounds on distributional convergence rates for high-dimensional vector- and matrix-valued functions of Markov chains. Methodologically, it integrates stochastic process theory, spectral analysis, and high-dimensional probability tools to derive high-probability consistency guarantees that match the asymptotic variance, achieving a Gaussian approximation rate of $O(T^{-1/4}log T)$ in convex distance. The key contributions are: (1) the first non-asymptotic, high-probability convergence guarantee for TD estimators under non-i.i.d. data—sharper than prior results; and (2) a rigorous theoretical foundation for uncertainty quantification and statistical inference in reinforcement learning.

Technology Category

• Machine Learning: Reinforcement Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Learning

Application Category

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Abstract

Markov chains are fundamental to statistical machine learning, underpinning key methodologies such as Markov Chain Monte Carlo (MCMC) sampling and temporal difference (TD) learning in reinforcement learning (RL). Given their widespread use, it is crucial to establish rigorous probabilistic guarantees on their convergence, uncertainty, and stability. In this work, we develop novel, high-dimensional concentration inequalities and Berry-Esseen bounds for vector- and matrix-valued functions of Markov chains, addressing key limitations in existing theoretical tools for handling dependent data. We leverage these results to analyze the TD learning algorithm, a widely used method for policy evaluation in RL. Our analysis yields a sharp high-probability consistency guarantee that matches the asymptotic variance up to logarithmic factors. Furthermore, we establish a $O(T^{-frac{1}{4}}log T)$ distributional convergence rate for the Gaussian approximation of the TD estimator, measured in convex distance. These findings provide new insights into statistical inference for RL algorithms, bridging the gaps between classical stochastic approximation theory and modern reinforcement learning applications.

Problem

• Uncertainty quantification for Markov chains
• High-dimensional concentration inequalities development
• Analysis of TD learning in reinforcement learning

Innovation

• High-dimensional concentration inequalities
• Berry-Esseen bounds for Markov chains
• Sharp high-probability consistency guarantee