Loop estimator for discounted values in Markov reward processes
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At the working heart of policy iteration algorithms commonly used and studied in the discounted setting of reinforcement learning, the policy evaluation step estimates the value of state with samples from a Markov reward process induced by following a Markov policy in a Markov decision process. We propose a simple and efficient estimator called loop estimator that exploits the regenerative structure of Markov reward processes without explicitly estimating a full model. Our method enjoys a space complexity of O(1) when estimating the value of a single positive recurrent state s unlike TD (with O(S)) or model-based methods (with O(S^2)). Moreover, the regenerative structure enables us to show, without relying on the generative model approach, that the estimator has an instance-dependent convergence rate of O(√(τ_s/T)) over steps T on a single sample path, where τ_s is the maximal expected hitting time to state s. In preliminary numerical experiments, the loop estimator outperforms model-free methods, such as TD(k), and is competitive with the model-based estimator.
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