  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.
