Local and Global Stability in Performative Reinforcement Learning

2026-09-06

AI Summary

本文探讨了表现型强化学习中的局部与全局稳定性问题，通过策略混合方法而非依赖难以验证的Lipschitz条件，提出了新的稳定性概念，并证明了在特定条件下Hedge动态可以达到稳定。

Abstract

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.

Problem

• performative reinforcement learning
• stability
• policy mixtures
• Lipschitz sensitivity
• multi-agent

Innovation

• performative reinforcement learning
• mixed policy stability
• weighted per-state Hedge dynamic
• bounded transition range assumption
• n-player performative Markov games