Continuous MDP Homomorphisms and Homomorphic Policy Gradient

Abstraction has been widely studied as a way to improve the efficiency and generalization of reinforcement learning algorithms. In this paper, we study abstraction in the continuous-control setting. We extend the definition of MDP homomorphisms to encompass continuous actions in continuous state spaces. We derive a policy gradient theorem on the abstract MDP, which allows us to leverage approximate symmetries of the environment for policy optimization. Based on this theorem, we propose an actor-critic algorithm that is able to learn the policy and the MDP homomorphism map simultaneously, using the lax bisimulation metric.