Policy-based methods have achieved remarkable success in solving challenging reinforcement learning problems. Among these methods, off-policy policy gradient methods are particularly important due to that they can benefit from off-policy data. However, these methods suffer from the high variance of the off-policy policy gradient (OPPG) estimator, which results in poor sample efficiency during training. In this paper, we propose an off-policy policy gradient method with the optimal action-dependent baseline (Off-OAB) to mitigate this variance issue. Specifically, this baseline maintains the OPPG estimator's unbiasedness while theoretically minimizing its variance. To enhance practical computational efficiency, we design an approximated version of this optimal baseline. Utilizing this approximation, our method (Off-OAB) aims to decrease the OPPG estimator's variance during policy optimization. We evaluate the proposed Off-OAB method on six representative tasks from OpenAI Gym and MuJoCo, where it demonstrably surpasses state-of-the-art methods on the majority of these tasks.

Policy gradient methods have enjoyed great success in deep reinforcement learning but suffer from high variance of gradient estimates. The high variance problem is particularly exasperated in problems with long horizons or high-dimensional action spaces. To mitigate this issue, we derive a bias-free action-dependent baseline for variance reduction which fully exploits the structural form of the stochastic policy itself and does not make any additional assumptions about the MDP. We demonstrate and quantify the benefit of the action-dependent baseline through both theoretical analysis as well as numerical results, including an analysis of the suboptimality of the optimal state-dependent baseline. The result is a computationally efficient policy gradient algorithm, which scales to high-dimensional control problems, as demonstrated by a synthetic 2000-dimensional target matching task. Our experimental results indicate that action-dependent baselines allow for faster learning on standard reinforcement learning benchmarks and high-dimensional hand manipulation and synthetic tasks. Finally, we show that the general idea of including additional information in baselines for improved variance reduction can be extended to partially observed and multi-agent tasks.