Uncertainty quantification for Markov chains with application to temporal difference learning

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.