Best Policy Identification in Linear MDPs
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We investigate the problem of best policy identification in discounted linear Markov Decision Processes in the fixed confidence setting under a generative model. We first derive an instance-specific lower bound on the expected number of samples required to identify an ε-optimal policy with probability 1-δ. The lower bound characterizes the optimal sampling rule as the solution of an intricate non-convex optimization program, but can be used as the starting point to devise simple and near-optimal sampling rules and algorithms. We devise such algorithms. One of these exhibits a sample complexity upper bounded by O(d/(ε+Δ)^2 (log(1/δ)+d)) where Δ denotes the minimum reward gap of sub-optimal actions and d is the dimension of the feature space. This upper bound holds in the moderate-confidence regime (i.e., for all δ), and matches existing minimax and gap-dependent lower bounds. We extend our algorithm to episodic linear MDPs.
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