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Game on Random Environment, Mean-Field Langevin System, and Neural Networks

Author

Listed:
  • Giovanni Conforti

    (Centre de Mathématiques Appliquées, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France)

  • Anna Kazeykina

    (Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91405 Orsay, France)

  • Zhenjie Ren

    (Ceremade, Université Paris Dauphine-PSL, 75016 Paris, France)

Abstract

In this paper, we study a class of games regularized by relative entropy where the players’ strategies are coupled through a random environment. Besides existence and uniqueness of equilibria for such games, we prove, under different sets of hypotheses that the marginal laws of the corresponding mean-field Langevin systems can converge toward the games’ equilibria. As an application, we show that dynamic games fall in this framework by considering the time horizon as environment. Concerning applications, our results allow analysis of stochastic gradient descent algorithms for deep neural networks in the context of supervised learning and for generative adversarial networks.

Suggested Citation

  • Giovanni Conforti & Anna Kazeykina & Zhenjie Ren, 2023. "Game on Random Environment, Mean-Field Langevin System, and Neural Networks," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 78-99, February.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:1:p:78-99
    DOI: 10.1287/moor.2022.1252
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