Offline Estimation of Controlled Markov Chains: Minimax Nonparametric Estimators and Sample Efficiency

Controlled Markov chains (CMCs) form the bedrock for model-based reinforcement learning. In this work, we consider the estimation of the transition probability matrices of a finite-state finite-control CMC using a fixed dataset, collected using a so-called logging policy, and develop minimax sample complexity bounds for nonparametric estimation of these transition probability matrices. Our results are general, and the statistical bounds depend on the logging policy through a natural mixing coefficient. We demonstrate an interesting trade-off between stronger assumptions on mixing versus requiring more samples to achieve a particular PAC-bound. We demonstrate the validity of our results under various examples, such as ergodic Markov chains, weakly ergodic inhomogeneous Markov chains, and controlled Markov chains with non-stationary Markov, episodic, and greedy controls. Lastly, we use these sample complexity bounds to establish concomitant ones for offline evaluation of stationary, Markov policies.