Approximation Benefits of Policy Gradient Methods with Aggregated States

Folklore suggests that policy gradient can be more robust to misspecification than its relative, approximate policy iteration. This paper studies the case of state-aggregation, where the state space is partitioned and either the policy or value function approximation is held constant over partitions. This paper shows a policy gradient method converges to a policy whose regret per-period is bounded by $\epsilon$, the largest difference between two elements of the state-action value function belonging to a common partition. With the same representation, both approximate policy iteration and approximate value iteration can produce policies whose per-period regret scales as $\epsilon/(1-\gamma)$, where $\gamma$ is a discount factor. Theoretical results synthesize recent analysis of policy gradient methods with insights of Van Roy (2006) into the critical role of state-relevance weights in approximate dynamic programming.