In the main text we present the TD and ETD algorithms for policy evaluation under linear function approximation, as a way to recognize the existing literature on emphatic algorithms [27]. We here present the derivation for policy evaluation under general function approximation. Following standard notation [41], capital letters for states, actions or rewards represent the random variable at time t (i.e. St is the random variable at time t) and lowercase letters represent their instantiation (i.e. St = sis the random variable St taking value sat time t).

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Policy gradient methods are widely adopted reinforcement learning algorithms for tasks with continuous action spaces. These methods succeeded in many application domains, however, because of their notorious sample inefficiency their use remains limited to problems where fast and accurate simulations are available. A common way to improve sample efficiency is to modify their objective function to be computable from off-policy samples without importance sampling. A well-established off-policy objective is the excursion objective. This work studies the difference between the excursion objective and the traditional on-policy objective, which we refer to as the on-off gap. We provide the first theoretical analysis showing conditions to reduce the on-off gap while establishing empirical evidence of shortfalls arising when these conditions are not met.

In the main text we present the TD and ETD algorithms for policy evaluation under linear function approximation, as a way to recognize the existing literature on emphatic algorithms [27]. We here present the derivation for policy evaluation under general function approximation. Following standard notation [41], capital letters for states, actions or rewards represent the random variable at time t (i.e. S

We propose a new objective, the counterfactual objective, unifying existing objectives for off-policy policy gradient algorithms in the continuing reinforcement learning (RL) setting. Compared to the commonly used excursion objective, which can be misleading about the performance of the target policy when deployed, our new objective better predicts such performance. We prove the Generalized Off-Policy Policy Gradient Theorem to compute the policy gradient of the counterfactual objective and use an emphatic approach to get an unbiased sample from this policy gradient, yielding the Generalized Off-Policy Actor-Critic (Geoff-PAC) algorithm. We demonstrate the merits of Geoff-PAC over existing algorithms in Mujoco robot simulation tasks, the first empirical success of emphatic algorithms in prevailing deep RL benchmarks.