  We propose the Bayes-UCBVI algorithm for reinforcement learning in tabular,
stage-dependent, episodic Markov decision process: a natural extension of the
Bayes-UCB algorithm by Kaufmann et al. (2012) for multi-armed bandits. Our
method uses the quantile of a Q-value function posterior as upper confidence
bound on the optimal Q-value function. For Bayes-UCBVI, we prove a regret bound
of order $\widetilde{O}(\sqrt{H^3SAT})$ where $H$ is the length of one episode,
$S$ is the number of states, $A$ the number of actions, $T$ the number of
episodes, that matches the lower-bound of $\Omega(\sqrt{H^3SAT})$ up to
poly-$\log$ terms in $H,S,A,T$ for a large enough $T$. To the best of our
knowledge, this is the first algorithm that obtains an optimal dependence on
the horizon $H$ (and $S$) without the need for an involved Bernstein-like bonus
or noise. Crucial to our analysis is a new fine-grained anti-concentration
bound for a weighted Dirichlet sum that can be of independent interest. We then
explain how Bayes-UCBVI can be easily extended beyond the tabular setting,
exhibiting a strong link between our algorithm and Bayesian bootstrap (Rubin,
1981).
