Refined Policy Improvement Bounds for MDPs.

The policy improvement bound on the difference of the discounted returns plays a crucial role in the theoretical justification of the trust-region policy optimization (TRPO) algorithm. The existing bound leads to a degenerate bound when the discount factor approaches one, making the applicability of TRPO and related algorithms questionable when the discount factor is close to one. We refine the results in \cite{Schulman2015, Achiam2017} and propose a novel bound that is "continuous" in the discount factor. In particular, our bound is applicable for MDPs with the long-run average rewards as well.