  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.

Machine Learning;Artificial Intelligence;Optimization and Control