We study the fundamental question of the sample complexity of learning a good policy in finite Markov decision processes (MDPs) when the data available for learning is obtained by following a logging policy that must be chosen without knowledge of the underlying MDP. Our main results show that the sample complexity, the minimum number of transitions necessary and sufficient to obtain a good policy, is an exponential function of the relevant quantities when the planning horizon $H$ is finite. In particular, we prove that the sample complexity of obtaining $epsilon$-optimal policies is at least $Omega(mathrm{A}^{min(mathrm{S}-1, H+1)})$ for $gamma$-discounted problems, where $mathrm{S}$ is the number of states, $mathrm{A}$ is the number of actions, and $H$ is the effective horizon defined as $H=lfloor tfrac{ln(1/epsilon)}{ln(1/gamma)} rfloor$; and it is at least $Omega(mathrm{A}^{min(mathrm{S}-1, H)}/varepsilon^2)$ for finite horizon problems, where $H$ is the planning horizon of the problem. This lower bound is essentially matched by an upper bound. For the average-reward setting we show that there is no algorithm finding $epsilon$-optimal policies with a finite amount of data.
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