#### Near-Optimal Time and Sample Complexities for Solving Markov Decision Processes with a Generative Model

Part of Advances in Neural Information Processing Systems 31 (NeurIPS 2018)

#### Authors

*Aaron Sidford, Mengdi Wang, Xian Wu, Lin Yang, Yinyu Ye*

#### Abstract

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 $\states$, actions $\actions$, 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 {\it both} the run time spent and number of sample taken is upper bounded by
\[
O\left[\frac{|\cS||\cA|}{(1-\gamma)^3 \epsilon^2} \log \left(\frac{|\cS||\cA|}{(1-\gamma)\delta \epsilon}
\right)
\log\left(\frac{1}{(1-\gamma)\epsilon}\right)\right] ~.
\]
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 \cite{azar2013minimax} 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.