  We propose UCBMQ, Upper Confidence Bound Momentum Q-learning, a new algorithm
for reinforcement learning in tabular and possibly stage-dependent, episodic
Markov decision process. UCBMQ is based on Q-learning where we add a momentum
term and rely on the principle of optimism in face of uncertainty to deal with
exploration. Our new technical ingredient of UCBMQ is the use of momentum to
correct the bias that Q-learning suffers while, at the same time, limiting the
impact it has on the second-order term of the regret. For UCBMQ , we are able
to guarantee a regret of at most $O(\sqrt{H^3SAT}+ H^4 S A )$ where $H$ is the
length of an episode, $S$ the number of states, $A$ the number of actions, $T$
the number of episodes and ignoring terms in poly$log(SAHT)$. Notably, UCBMQ is
the first algorithm that simultaneously matches the lower bound of
$\Omega(\sqrt{H^3SAT})$ for large enough $T$ and has a second-order term (with
respect to the horizon $T$) that scales only linearly with the number of states
$S$.
