  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.
