Stability of Underdamped Langevin Dynamics in Distributional Minimax Optimization

Underdamped Langevin methods have received increasing attention in mean-field optimization, motivated by their convergence and discretization properties. In this paper, we show that for distributional minimax optimization, coupling the two players can destabilize underdamped Langevin dynamics. For a bilinear game, we derive an exact friction–coupling threshold separating exponentially stable and unstable underdamped dynamics, while the overdamped counterpart remains stable at every coupling strength. We further show that this instability persists for a bounded smooth interaction. To avoid simultaneous cross-player coupling, we introduce an anchored scheme that freezes the current strategies while running underdamped Langevin dynamics toward their Gibbs responses, and updates the strategies between these runs. For a class of convex-concave functionals of probability laws in the mean-field setting, we establish last-iterate convergence of the proposed scheme, with an $O(1/K)$ decay of the Nikaido–Isoda error. Lastly, we provide a finite-particle, discrete-time implementation with a last-iterate convergence guarantee. More broadly, our results provide a principled framework for understanding and stabilizing underdamped dynamics in distributional minimax optimization.