We examine online safe multi-agent reinforcement learning using constrained Markov games in which agents compete by maximizing their expected total rewards under a constraint on expected total utilities. Our focus is confined to an episodic two-player zero-sum constrained Markov game with independent transition functions that are unknown to agents, adversarial reward functions, and stochastic utility functions. For such a Markov game, we employ an approach based on the occupancy measure to formulate it as an online constrained saddle-point problem with an explicit constraint. We extend the Lagrange multiplier method in constrained optimization to handle the constraint by creating a generalized Lagrangian with minimax decision primal variables and a dual variable. Next, we develop an upper confidence reinforcement learning algorithm to solve this Lagrangian problem while balancing exploration and exploitation. Our algorithm updates the minimax decision primal variables via online mirror descent and the dual variable via projected gradient step and we prove that it enjoys sublinear rate $ O((|X|+|Y|) L \sqrt{T(|A|+|B|)}))$ for both regret and constraint violation after playing $T$ episodes of the game. Here, $L$ is the horizon of each episode, $(|X|,|A|)$ and $(|Y|,|B|)$ are the state/action space sizes of the min-player and the max-player, respectively. To the best of our knowledge, we provide the first provably efficient online safe reinforcement learning algorithm in constrained Markov games.

本文提出一种使用基于占用测度的拉格朗日优化方法来解决约束马尔可夫博弈的在线安全强化学习算法，经更新的minimax决策原始变量和双重变量，达到亚线性后悔率和约束违规率，实现对马尔可夫博弈的高效学习。

面向安全多智体强化学习的可证明高效广义拉格朗日策略优化