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An implicit gradient-descent procedure for minimax problems

Author

Listed:
  • Montacer Essid

    (Courant Institute of Mathematical Sciences)

  • Esteban G. Tabak

    (Courant Institute of Mathematical Sciences)

  • Giulio Trigila

    (Baruch College)

Abstract

A game theory inspired methodology is proposed for finding a function’s saddle points. While explicit descent methods are known to have severe convergence issues, implicit methods are natural in an adversarial setting, as they take the other player’s optimal strategy into account. The implicit scheme proposed has an adaptive learning rate that makes it transition to Newton’s method in the neighborhood of saddle points. Convergence is shown through local analysis and through numerical examples in optimal transport and linear programming. An ad-hoc quasi-Newton method is developed for high dimensional problems, for which the inversion of the Hessian of the objective function may entail a high computational cost.

Suggested Citation

  • Montacer Essid & Esteban G. Tabak & Giulio Trigila, 2023. "An implicit gradient-descent procedure for minimax problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 97(1), pages 57-89, February.
    • Handle: RePEc:spr:mathme:v:97:y:2023:i:1:d:10.1007_s00186-022-00805-w
    DOI: 10.1007/s00186-022-00805-w
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