Oscillating solutions to the mean-field Langevin descent-ascent flow

Jean-Christophe Mourrat, Loucas Pillaud-Vivien

We study a particle approximation of the mean-field Langevin descent-ascent dynamics on ℝ², where the payoff function is shaped as a double well in each coordinate. When the coupling α is sufficiently strong and the entropic regularization ε sufficiently small, the deterministic dynamics admits a limit cycle, and we show that the mean-field dynamics stay close to this cyclic behavior and do not converge.

Below we display phase portrait animations of 500 interacting particles (α = 1.5) for two noise levels. For ε = 0.25 (left), the particles concentrate around the limit cycle of the deterministic system (black curve). For ε = 0.5 (right), the noise dominates and the particles spread out toward a stationary measure resembling a mixture of four Gaussians.