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A Hölderian backtracking method for min-max and min-min problems

Author

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  • Bolte, Jérôme
  • Glaudin, Lilian
  • Pauwels, Edouard
  • Serrurier, Matthieu

Abstract

We present a new algorithm to solve min-max or min-min problems out of the convex world. We use rigidity assumptions, ubiquitous in learning, making our method applicable to many optimization problems. Our approach takes advantage of hidden regularity properties and allows us to devise a simple algorithm of ridge type. An original feature of our method is to come with automatic step size adaptation which departs from the usual overly cautious backtracking methods. In a general framework, we provide convergence theoretical guarantees and rates. We apply our findings on simple GAN problems obtaining promising numerical results

Suggested Citation

  • Bolte, Jérôme & Glaudin, Lilian & Pauwels, Edouard & Serrurier, Matthieu, 2021. "A Hölderian backtracking method for min-max and min-min problems," TSE Working Papers 21-1243, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:125888
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    References listed on IDEAS

    as
    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Bolte, Jérôme & Castera, Camille & Pauwels, Edouard & Févotte, Cédric, 2019. "An Inertial Newton Algorithm for Deep Learning," TSE Working Papers 19-1043, Toulouse School of Economics (TSE).
    3. NUNES GRAPIGLIA Geovani, & NESTEROV Yurii,, 2019. "Tensor methods for minimizing convex functions with Hölder continuous higher-order derivatives," LIDAM Discussion Papers CORE 2019028, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Jérôme Bolte & Shoham Sabach & Marc Teboulle, 2018. "Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1210-1232, November.
    6. R. T. Rockafellar, 1981. "Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization," Mathematics of Operations Research, INFORMS, vol. 6(3), pages 424-436, August.
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