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Perturbed Iterate SGD for Lipschitz Continuous Loss Functions

Author

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  • Michael R. Metel

    (Huawei Noah’s Ark Lab)

  • Akiko Takeda

    (The University of Tokyo
    RIKEN Center for Advanced Intelligence Project)

Abstract

This paper presents an extension of stochastic gradient descent for the minimization of Lipschitz continuous loss functions. Our motivation is for use in non-smooth non-convex stochastic optimization problems, which are frequently encountered in applications such as machine learning. Using the Clarke $$\epsilon $$ ϵ -subdifferential, we prove the non-asymptotic convergence to an approximate stationary point in expectation for the proposed method. From this result, a method with non-asymptotic convergence with high probability, as well as a method with asymptotic convergence to a Clarke stationary point almost surely are developed. Our results hold under the assumption that the stochastic loss function is a Carathéodory function which is almost everywhere Lipschitz continuous in the decision variables. To the best of our knowledge, this is the first non-asymptotic convergence analysis under these minimal assumptions.

Suggested Citation

  • Michael R. Metel & Akiko Takeda, 2022. "Perturbed Iterate SGD for Lipschitz Continuous Loss Functions," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 504-547, November.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:2:d:10.1007_s10957-022-02093-0
    DOI: 10.1007/s10957-022-02093-0
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    References listed on IDEAS

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    1. J. V. Burke & A. S. Lewis & M. L. Overton, 2002. "Approximating Subdifferentials by Random Sampling of Gradients," Mathematics of Operations Research, INFORMS, vol. 27(3), pages 567-584, August.
    2. Yurii NESTEROV & Vladimir SPOKOINY, 2017. "Random gradient-free minimization of convex functions," LIDAM Reprints CORE 2851, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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