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

TL;DR

This paper provides a counterexample demonstrating that the mean-field Langevin descent-ascent flow can exhibit non-convergent cyclic behavior under certain conditions involving double well payoff functions and strong coupling.

Contribution

It shows that the mean-field Langevin descent-ascent flow does not always converge, especially with specific payoff functions and parameter regimes, challenging previous assumptions.

Findings

The mean-field dynamics can admit a limit cycle in certain settings.

Strong coupling and small entropic regularization lead to cyclic, non-convergent behavior.

The paper constructs a specific counterexample with double well functions.

Abstract

We present a counterexample to the statement of convergence of the mean-field Langevin descent-ascent flow on $\mathbb{R}^2$. We consider payoff functions that are shaped as a double well in each coordinate, and for which the deterministic dynamics admits a limit cycle. When the coupling between the two coordinates is sufficiently strong and the entropic regularization sufficiently small, we show that the mean-field dynamics remains close to this cyclic behavior, and in particular, does not converge.