Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model

In this paper we consider the problem of computing an $\epsilon$-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 $\gamma\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $\epsilon$-optimal policy with probability $1 - \delta$ where \emph{both} the time spent and number of sample taken are upper bounded by $$O\left[\frac{|S||A|}{(1-\gamma)^3 \epsilon^2} \log \left(\frac{|S||A|}{(1-\gamma)\delta \epsilon} \right) \log\left(\frac{1}{(1-\gamma)\epsilon}\right)\right] ~.$$ For fixed values of $\epsilon$, this improves upon the previous best known bounds by a factor of $(1 - \gamma)^{-1}$ and matches the sample complexity lower bounds proved in Azar et al. (2013) up to logarithmic factors. We also extend our method to computing $\epsilon$-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.