  We consider reinforcement learning in an environment modeled by an episodic,
finite, stage-dependent Markov decision process of horizon $H$ with $S$ states,
and $A$ actions. The performance of an agent is measured by the regret after
interacting with the environment for $T$ episodes. We propose an optimistic
posterior sampling algorithm for reinforcement learning (OPSRL), a simple
variant of posterior sampling that only needs a number of posterior samples
logarithmic in $H$, $S$, $A$, and $T$ per state-action pair. For OPSRL we
guarantee a high-probability regret bound of order at most
$\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ ignoring $\text{poly}\log(HSAT)$
terms. The key novel technical ingredient is a new sharp anti-concentration
inequality for linear forms which may be of independent interest. Specifically,
we extend the normal approximation-based lower bound for Beta distributions by
Alfers and Dinges [1984] to Dirichlet distributions. Our bound matches the
lower bound of order $\Omega(\sqrt{H^3SAT})$, thereby answering the open
problems raised by Agrawal and Jia [2017b] for the episodic setting.
