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Minimax PAC bounds on the sample complexity of reinforcement learning with a generative model

Mohammad Gheshlaghi Azar 1 Rémi Munos 2 Hilbert Kappen 1
2 SEQUEL - Sequential Learning
LIFL - Laboratoire d'Informatique Fondamentale de Lille, Inria Lille - Nord Europe, LAGIS - Laboratoire d'Automatique, Génie Informatique et Signal
Abstract : We consider the problem of learning the optimal action-value function in discounted-reward Markov decision processes (MDPs). We prove new PAC bounds on the sample-complexity of two well-known model-based reinforcement learning (RL) algorithms in the presence of a generative model of the MDP: value iteration and policy iteration. The first result indicates that for an MDP with $N$ state-action pairs and the discount factor γin[0, 1) only $O(N log(N/δ)/ [(1 - γ)3 \epsilon^2])$ state-transition samples are required to find an $\epsilon$-optimal estimation of the action-value function with the probability (w.p.) 1-δ. Further, we prove that, for small values of $\epsilon$, an order of $O(N log(N/δ)/ [(1 - γ)3 \epsilon^2])$ samples is required to find an $\epsilon$ -optimal policy w.p. 1-δ. We also prove a matching lower bound of $\Omega(N log(N/δ)/ [(1 - γ)3\epsilon2])$ on the sample complexity of estimating the optimal action-value function. To the best of our knowledge, this is the first minimax result on the sample complexity of RL: The upper bound matches the lower bound interms of $N$ , $\epsilon$, δ and 1/(1 -γ) up to a constant factor. Also, both our lower bound and upper bound improve on the state-of-the-art in terms of their dependence on 1/(1-γ).
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Mohammad Gheshlaghi Azar, Rémi Munos, Hilbert Kappen. Minimax PAC bounds on the sample complexity of reinforcement learning with a generative model. Machine Learning, Springer Verlag, 2013, 91 (3), pp.325-349. ⟨10.1007/s10994-013-5368-1⟩. ⟨hal-00831875⟩

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