Policy gradient methods are widely used for control in reinforcement learning, particularly for the continuous action setting. There have been a host of theoretically sound algorithms proposed for the on-policy setting, due to the existence of the policy gradient theorem which provides a simplified form for the gradient. In off-policy learning, however, where the behaviour policy is not necessarily attempting to learn and follow the optimal policy for the given task, the existence of such a theorem has been elusive. In this work, we solve this open problem by providing the first off-policy policy gradient theorem. The key to the derivation is the use of $emphatic$ $weightings$. We develop a new actor-critic algorithm$unicode{x2014}$called Actor Critic with Emphatic weightings (ACE)$unicode{x2014}$that approximates the simplified gradients provided by the theorem. We demonstrate in a simple counterexample that previous off-policy policy gradient methods$unicode{x2014}$particularly OffPAC and DPG$unicode{x2014}$converge to the wrong solution whereas ACE finds the optimal solution.
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