We develop a parameterized Primal-Dual $\pi$ Learning method based on deep neural networks for Markov decision process with large state space and off-policy reinforcement learning. In contrast to the popular Q-learning and actor-critic methods that are based on successive approximations to the nonlinear Bellman equation, our method makes primal-dual updates to the policy and value functions utilizing the fundamental linear Bellman duality. Naive parametrization of the primal-dual $\pi$ learning method using deep neural networks would encounter two major challenges: (1) each update requires computing a probability distribution over the state space and is intractable; (2) the iterates are unstable since the parameterized Lagrangian function is no longer linear. We address these challenges by proposing a relaxed Lagrangian formulation with a regularization penalty using the advantage function. We show that the dual policy update step in our method is equivalent to the policy gradient update in the actor-critic method in some special case, while the value updates differ substantially. The main advantage of the primal-dual $\pi$ learning method lies in that the value and policy updates are closely coupled together using the Bellman duality and therefore more informative. Experiments on a simple cart-pole problem show that the algorithm significantly outperforms the one-step temporal-difference actor-critic method, which is the most relevant benchmark method to compare with. We believe that the primal-dual updates to the value and policy functions would expedite the learning process. The proposed methods might open a door to more efficient algorithms and sharper theoretical analysis.

基于深度神经网络的参数Primal-Dual pi学习方法，旨在解决马尔可夫决策过程中状态空间大和策略离线学习问题，通过基本线性Bellman方法对价值和策略函数进行原始对偶更新，从而更加有效地进行价值和策略更新，在与同类方法比较的测试中表现明显优于最相关的基准方法

深度原始-对偶强化学习：利用贝尔曼对偶加速演员-评论家算法