## Abstract

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the Mean-Field Gradient Descent Ascent (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable finite scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

Original language | English (US) |
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Pages (from-to) | 22790-22811 |

Number of pages | 22 |

Journal | Proceedings of Machine Learning Research |

Volume | 202 |

State | Published - 2023 |

Externally published | Yes |

Event | 40th International Conference on Machine Learning, ICML 2023 - Honolulu, United States Duration: Jul 23 2023 → Jul 29 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Proceedings of Machine Learning Research. All rights reserved.