In this paper we consider the problem of learning an $\epsilon$-optimal policy for a discounted Markov Decision Process (MDP). Given an MDP with $S$ states, $A$ actions, the discount factor $\gamma \in (0,1)$, and an approximation threshold $\epsilon > 0$, we provide a model-free algorithm to learn an $\epsilon$-optimal policy with sample complexity $\tilde{ O}(\frac{ SA\ln(1/p)}{ \epsilon^2(1-\gamma)^{ 5.5}})$ (where the notation $\tilde{ O}(\cdot)$ hides poly-logarithmic factors of $S,A,1/(1-\gamma)$, and $1/\epsilon$) and success probability $(1-p)$. For small enough $\epsilon$, we show an improved algorithm with sample complexity $\tilde{ O}(\frac{ SA\ln(1/p)}{ \epsilon^2(1-\gamma)^{ 3}})$. While the first bound improves upon all known model-free algorithms and model-based ones with tight dependence on $S$, our second algorithm beats all known sample complexity bounds and matches the information theoretic lower bound up to logarithmic factors.
Categories: cs.LG stat.ML