Analysis of On-policy Policy Gradient Methods under the Distribution Mismatch

Policy gradient methods are one of the most successful approaches for solving challenging reinforcement learning problems. Despite their empirical successes, many state-of-the-art policy gradient algorithms for discounted problems deviate from the theoretical policy gradient theorem due to the existence of a distribution mismatch. In this work, we analyze the impact of this mismatch on policy gradient methods. Specifically, we first show that in the case of tabular parameterizations, the biased gradient induced by the mismatch still yields a valid first-order characterization of global optimality. Then, we extend this analysis to more general parameterizations by deriving explicit bounds on both the state distribution mismatch and the resulting gradient mismatch in episodic and continuing MDPs, which are shown to vanish at least linearly as the discount factor approaches one. Building on these bounds, we further establish guarantees for the biased policy gradient iterates, showing that they approach approximate stationary points with respect to the exact gradient, with asymptotic residuals depending on the discount factor. Our findings offer insights into the robustness of policy gradient methods as well as the gap between theoretical foundations and practical implementations.