Edge of Stability Occurs in Minimax Optimization

The Edge of Stability (EoS) phenomenon is widely observed in minimization, where gradient methods continue to decrease the loss even while remaining near their stability thresholds, a behavior associated with a self-stabilization mechanism. Yet an analogous phenomenon has not been clearly identified in minimax optimization. In this paper, we formulate EoS for minimax optimization, demonstrate that it indeed occurs, and characterize the self-stabilization mechanism underlying it. We first define EoS for minimax optimization in terms of possibly complex dominant Jacobian eigenvalues reaching the boundary of the stability region. Using two-player zero-sum games, we observe minimax EoS for both Extragradient (EG) and Optimistic Gradient Descent Ascent (OGDA). However, when the dominant eigenvalues lie near the real axis, OGDA still exhibits EoS whereas EG does not. Finally, we generalize the self-stabilization mechanism to minimax methods and show how it explains these behaviors.