  We derive a novel asymptotic problem-dependent lower-bound for regret
minimization in finite-horizon tabular Markov Decision Processes (MDPs). While,
similar to prior work (e.g., for ergodic MDPs), the lower-bound is the solution
to an optimization problem, our derivation reveals the need for an additional
constraint on the visitation distribution over state-action pairs that
explicitly accounts for the dynamics of the MDP. We provide a characterization
of our lower-bound through a series of examples illustrating how different MDPs
may have significantly different complexity. 1) We first consider a "difficult"
MDP instance, where the novel constraint based on the dynamics leads to a
larger lower-bound (i.e., a larger regret) compared to the classical analysis.
2) We then show that our lower-bound recovers results previously derived for
specific MDP instances. 3) Finally, we show that, in certain "simple" MDPs, the
lower bound is considerably smaller than in the general case and it does not
scale with the minimum action gap at all. We show that this last result is
attainable (up to $poly(H)$ terms, where $H$ is the horizon) by providing a
regret upper-bound based on policy gaps for an optimistic algorithm.
