Abstract: This paper presents the first non-asymptotic result showing a model-free algorithm can achieve logarithmic cumulative regret for episodic tabular reinforcement learning if there exists a strictly positive sub-optimality gap. We prove that the optimistic Q-learning studied in [Jin et al. 2018] enjoys a {\mathcal{O}}\!\left(\frac{SA\cdot \mathrm{poly}\left(H\right)}{\Delta_{\min}}\log\left(SAT\right)\right) cumulative regret bound where S is the number of states, A is the number of actions, H is the planning horizon, T is the total number of steps, and \Delta_{\min} is the minimum sub-optimality gap of the optimal Q-function. This bound matches the information theoretical lower bound in terms of S,A,T up to a \log\left(SA\right) factor. We further extend our analysis to the discounted setting and obtain a similar logarithmic cumulative regret bound.