Personalized Federated Reinforcement Learning via Model-Agnostic Meta-Learning: Convergence of Exact and Hessian-Free Meta-Policy Gradients

We study personalized federated reinforcement learning, in which $n$ agents, each acting in its own Markov decision process, collaborate through a server to learn a shared MAML-style policy initialization that becomes effective for an individual agent once that agent adapts it with a single local policy-gradient step. We propose Per-FedAvg-PG, in which agents take $τ$ local stochastic meta-policy-gradient steps between communication rounds, and prove that it reaches an $\varepsilon$-approximate first-order stationary point of the personalized objective in $K=\mathcal O(\varepsilon^{-3/2})$ rounds with $τ=Θ(\varepsilon^{-1/2})$ local steps. The analysis rests on a structural feature of the reinforcement learning setting: under standard policy-class regularity, the per-agent objectives have uniformly bounded gradients and Hessians with explicit constants, so the bounded-gradient and bounded-heterogeneity conditions imposed by the supervised theory hold automatically and no separate heterogeneity assumption is needed. The exact meta-gradient requires the inner-loop policy Hessian, which our experiments identify as the practical bottleneck. We therefore analyze the Hessian-free variant, bound its bias, and exhibit fixed points at which the meta-gradient is nonzero and of order $α$, showing that the resulting stationarity floor is a property of the method rather than of the bound. Experiments on tabular and neural navigation confirm the predicted behavior and show transfer to unseen agents at an order of magnitude lower sample cost than independent training. Together these results identify the adaptation step size as a tunable personalization knob and the curvature estimate as the quantity that governs whether exact meta-gradients are affordable.