Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model
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We investigate the sample efficiency of reinforcement learning in a γ-discounted infinite-horizon Markov decision process (MDP) with state space S and action space A, assuming access to a generative model. Despite a number of prior work tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, prior results suffer from a sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least |S||A|/(1-γ)^2 (up to some log factor). The current paper overcomes this barrier by certifying the minimax optimality of model-based reinforcement learning as soon as the sample size exceeds the order of |S||A|/1-γ (modulo some log factor). More specifically, a perturbed model-based planning algorithm provably finds an ε-optimal policy with an order of |S||A| /(1-γ)^3ε^2log|S||A|/(1-γ)ε samples for any ε∈ (0, 1/1-γ]. Along the way, we derive improved (instance-dependent) guarantees for model-based policy evaluation. To the best of our knowledge, this work provides the first minimax-optimal guarantee in a generative model that accommodates the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically impossible).
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