  For the problem of task-agnostic reinforcement learning (RL), an agent first
collects samples from an unknown environment without the supervision of reward
signals, then is revealed with a reward and is asked to compute a corresponding
near-optimal policy. Existing approaches mainly concern the worst-case
scenarios, in which no structural information of the reward/transition-dynamics
is utilized. Therefore the best sample upper bound is
$\propto\widetilde{\mathcal{O}}(1/\epsilon^2)$, where $\epsilon>0$ is the
target accuracy of the obtained policy, and can be overly pessimistic. To
tackle this issue, we provide an efficient algorithm that utilizes a gap
parameter, $\rho>0$, to reduce the amount of exploration. In particular, for an
unknown finite-horizon Markov decision process, the algorithm takes only
$\widetilde{\mathcal{O}} (1/\epsilon \cdot (H^3SA / \rho + H^4 S^2 A) )$
episodes of exploration, and is able to obtain an $\epsilon$-optimal policy for
a post-revealed reward with sub-optimality gap at least $\rho$, where $S$ is
the number of states, $A$ is the number of actions, and $H$ is the length of
the horizon, obtaining a nearly \emph{quadratic saving} in terms of $\epsilon$.
We show that, information-theoretically, this bound is nearly tight for $\rho <
\Theta(1/(HS))$ and $H>1$. We further show that
$\propto\widetilde{\mathcal{O}}(1)$ sample bound is possible for $H=1$ (i.e.,
multi-armed bandit) or with a sampling simulator, establishing a stark
separation between those settings and the RL setting.
