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Mean-Field Control (MFC) has recently been proven to be a scalable tool toapproximately solve large-scale multi-agent reinforcement learning (MARL)problems. However, these studies are typically limited to unconstrainedcumulative reward maximization framework. In this paper, we show that one canuse the MFC approach to approximate the MARL problem even in the presence ofconstraints. Specifically, we prove that, an $N$-agent constrained MARLproblem, with state, and action spaces of each individual agents being of sizes$|\mathcal{X}|$, and $|\mathcal{U}|$ respectively, can be approximated by anassociated constrained MFC problem with an error, $e\triangleq\mathcal{O}\left([\sqrt{|\mathcal{X}|}+\sqrt{|\mathcal{U}|}]/\sqrt{N}\right)$.In a special case where the reward, cost, and state transition functions areindependent of the action distribution of the population, we prove that theerror can be improved to $e=\mathcal{O}(\sqrt{|\mathcal{X}|}/\sqrt{N})$. Also,we provide a Natural Policy Gradient based algorithm and prove that it cansolve the constrained MARL problem within an error of $\mathcal{O}(e)$ with asample complexity of $\mathcal{O}(e^{-6})$.

Mean-Field Approximation of Cooperative Constrained Multi-Agent Reinforcement Learning (CMARL)