Title: Learning in Circles: Rotational Dynamics in Competitive Reinforcement Learning
Primary area: reinforcement learning
Keywords: Game theory, Optimization, Multi-agent Reinforcement learning

Abstract: Optimization in competitive reinforcement learning (RL) differs from standard minimization. Actor–critic methods, in single- and multi-agent (MARL) settings, involve coupled objectives, so optimizing them jointly requires finding an equilibrium rather than performing independent descent. Through operator-theoretic viewpoint, we show that actor–critic models inherently exhibit rotational dynamics during learning, cycling around equilibria, thereby explaining in part the instability often observed in practice.  Through the variational inequality (VI) framework for studying equilibrium seeking problems, we adopt the Lookahead method for VIs, which suppresses these rotations in actor–critic RL. Building on this, we introduce *Lookahead-(MA)RL (LA-(MA)RL)* to efficiently mitigate rotational dynamics. Across classical two-player games and multi-agent benchmarks, including *Rock--paper--scissors*, *Matching pennies*, and *Multi-Agent Particle environments*, LA-MARL consistently improves convergence and stability. Our results highlight optimization as a critical yet underexplored lever in RL: by rethinking the equilibrium-seeking dynamics, one can achieve substantial stability and performance gains.