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Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model
Aaron Sidford, Mengdi Wang, Xian Wu, Lin F. Yang, Yinyu Ye
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In this paper we consider the problem of computing an ϵ-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in O(1) time. Given such a DMDP with states S, actions A, discount factor γ∈(0,1), and rewards in range [0, 1] we provide an algorithm which computes an ϵ-optimal policy with probability 1 - δ where \emph{both} the time spent and number of sample taken are upper bounded by O[(|S||A|)/((1-γ)³ ϵ²) log((|S||A|)/((1-γ)δϵ) ) log(1/((1-γ)ϵ))] . For fixed values of ϵ, this improves upon the previous best known bounds by a factor of (1 - γ)⁻¹ and matches the sample complexity lower bounds proved in Azar et al. (2013) up to logarithmic factors. We also extend our method to computing ϵ-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.
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