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 staying near the boundary of classical stability. In contrast, an analogous phenomenon has not been clearly identified in minimax optimization. In this paper, we show that EoS indeed occurs in minimax optimization. To characterize EoS beyond minimization, we formulate EoS in terms of the stability region of minimax dynamics. This formulation naturally captures a rotational form of EoS associated with complex dominant eigenvalues of the Jacobian, which does not appear in minimization. To investigate this rotational EoS, we consider a two-stage variant of the Rock-Paper-Scissors game as a simple testbed and empirically demonstrate EoS for both Extragradient (EG) and Optimistic Gradient Descent Ascent (OGDA). However, when the dominant eigenvalue is near the real axis, corresponding to the non-rotational case as in minimization, we show that OGDA still exhibits EoS, whereas EG does not.