We discuss the problem of decentralized multi-agent reinforcement learning (MARL) in this work. In our setting, the global state, action, and reward are assumed to be fully observable, while the local policy is protected as privacy by each agent, and thus cannot be shared with others. There is a communication graph, among which the agents can exchange information with their neighbors. The agents make individual decisions and cooperate to reach a higher accumulated reward. Towards this end, we first propose a decentralized actor-critic (AC) setting. Then, the policy evaluation and policy improvement algorithms are designed for discrete and continuous state-action-space Markov Decision Process (MDP) respectively. Furthermore, convergence analysis is given under the discrete-space case, which guarantees that the policy will be reinforced by alternating between the processes of policy evaluation and policy improvement. In order to validate the effectiveness of algorithms, we design experiments and compare them with previous algorithms, e.g., Q-learning \cite{watkins1992q} and MADDPG \cite{lowe2017multi}. The results show that our algorithms perform better from the aspects of both learning speed and final performance. Moreover, the algorithms can be executed in an off-policy manner, which greatly improves the data efficiency compared with on-policy algorithms.
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