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Nonsmooth Nonconvex Stochastic Heavy Ball

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  • Tam Le

    (University of Toulouse)

Abstract

Motivated by the conspicuous use of momentum-based algorithms in deep learning, we study a nonsmooth nonconvex stochastic heavy ball method and show its convergence. Our approach builds upon semialgebraic (definable) assumptions commonly met in practical situations and combines a nonsmooth calculus with a differential inclusion method. Additionally, we provide general conditions for the sample distribution to ensure the convergence of the objective function. Our results are general enough to justify the use of subgradient sampling in modern implementations that heuristically apply rules of differential calculus on nonsmooth functions, such as backpropagation or implicit differentiation. As for the stochastic subgradient method, our analysis highlights that subgradient sampling can make the stochastic heavy ball method converge to artificial critical points. Thanks to the semialgebraic setting, we address this concern showing that these artifacts are almost surely avoided when initializations are randomized, leading the method to converge to Clarke critical points.

Suggested Citation

  • Tam Le, 2024. "Nonsmooth Nonconvex Stochastic Heavy Ball," Journal of Optimization Theory and Applications, Springer, vol. 201(2), pages 699-719, May.
    • Handle: RePEc:spr:joptap:v:201:y:2024:i:2:d:10.1007_s10957-024-02408-3
    DOI: 10.1007/s10957-024-02408-3
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    References listed on IDEAS

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    1. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    2. Bolte, Jérôme & Le, Tam & Pauwels, Edouard, 2022. "Subgradient sampling for nonsmooth nonconvex minimization," TSE Working Papers 22-1310, Toulouse School of Economics (TSE).
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    Cited by:

    1. Kuangyu Ding & Kim-Chuan Toh, 2025. "Stochastic Bregman Subgradient Methods for Nonsmooth Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-36, September.

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