  This paper establishes that optimistic algorithms attain gap-dependent and
non-asymptotic logarithmic regret for episodic MDPs. In contrast to prior work,
our bounds do not suffer a dependence on diameter-like quantities or
ergodicity, and smoothly interpolate between the gap dependent
logarithmic-regret, and the $\widetilde{\mathcal{O}}(\sqrt{HSAT})$-minimax
rate. The key technique in our analysis is a novel "clipped" regret
decomposition which applies to a broad family of recent optimistic algorithms
for episodic MDPs.
