Episodic reinforcement learning and contextual bandits are two widely studied sequential decision-making problems. Episodic reinforcement learning generalizes contextual bandits and is often perceived to be more difficult due to long planning horizon and unknown state-dependent transitions. The current paper shows that the long planning horizon and the unknown state-dependent transitions (at most) pose little additional difficulty on sample complexity. We consider the episodic reinforcement learning with $S$ states, $A$ actions, planning horizon $H$, total reward bounded by $1$, and the agent plays for $K$ episodes. We propose a new algorithm, \textbf{ M}onotonic \textbf{ V}alue \textbf{ P}ropagation (MVP), which relies on a new Bernstein-type bonus. Compared to existing bonus constructions, the new bonus is tighter since it is based on a well-designed monotonic value function. In particular, the \emph{ constants} in the bonus should be subtly setting to ensure optimism and monotonicity. We show MVP enjoys an $O\left(\left(\sqrt{ SAK} + S^2A\right) \poly\log \left(SAHK\right)\right)$ regret, approaching the $\Omega\left(\sqrt{ SAK}\right)$ lower bound of \emph{ contextual bandits} up to logarithmic terms. Notably, this result 1) \emph{ exponentially} improves the state-of-the-art polynomial-time algorithms by Dann et al. [2019] and Zanette et al. [2019] in terms of the dependency on $H$, and 2) \emph{ exponentially} improves the running time in [Wang et al. 2020] and significantly improves the dependency on $S$, $A$ and $K$ in sample complexity.
Categories: cs.LG stat.ML