  We devise and analyze algorithms for the empirical policy evaluation problem
in reinforcement learning. Our algorithms explore backward from high-cost
states to find high-value ones, in contrast to forward approaches that work
forward from all states. While several papers have demonstrated the utility of
backward exploration empirically, we conduct rigorous analyses which show that
our algorithms can reduce average-case sample complexity from $O(S \log S)$ to
as low as $O(\log S)$.
