Local and Global Stability in Performative Reinforcement Learning
In performative reinforcement learning the deployed policy shapes the environment that generates the learner's future data, and the natural solution concept is a performatively stable policy that is optimal in the environment it induces. Existing convergence guarantees rely on Lipschitz sensitivity assumptions on the environment map $\pi \mapsto (P_\pi, r_\pi)$, which are hard to verify and fail in settings such as multi-agent best-response dynamics. We instead study stability for mixtures of policies, and show that the resulting picture is fundamentally different from performative prediction, where randomization removes the need for any sensitivity assumption. We distinguish local mixed stability, an occupancy-weighted first-order relaxation that we show is equivalent to stationarity, from global mixed stability, which certifies against arbitrary deviating policies. Our first result is that a weighted per-state Hedge dynamic drives the local stability gap to zero at an $O(1/\sqrt{T})$ rate for an arbitrary, possibly discontinuous, environment map, both with exact and with trajectory feedback. The two notions genuinely differ: we exhibit an instance where local stability is achieved exactly but every mixture has global stability gap bounded away from zero. For global stability we introduce a bounded transition range assumption, strictly weaker than Lipschitz sensitivity, under which unweighted per-state Hedge converges up to a floor of $O(\gamma\epsilon_P/(1-\gamma)^3)$, and we prove a matching-in-$\epsilon_P$ lower bound of $\Omega(\gamma\epsilon_P/(1-\gamma))$ under trajectory feedback, so this floor is unavoidable. Finally, we extend both notions to $n$-player performative Markov games, obtaining local stability with no assumption on the joint environment map or game structure, and global stability for performative Markov potential games.