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The “Black-Box” Optimization Problem: Zero-Order Accelerated Stochastic Method via Kernel Approximation

Author

Listed:
  • Aleksandr Lobanov

    (9 Institutskiy per
    Skolkovo Institute of Science and Technology
    ISP RAS Research Center for Trusted Artificial Intelligence)

  • Nail Bashirov

    (9 Institutskiy per
    Institute for Information Transmission Problems)

  • Alexander Gasnikov

    (9 Institutskiy per
    ISP RAS Research Center for Trusted Artificial Intelligence
    Innopolis University)

Abstract

In this paper, we study the standard formulation of an optimization problem when the computation of gradient is not available. Such a problem can be classified as a “black box” optimization problem, since the oracle returns only the value of the objective function at the requested point, possibly with some stochastic noise. Assuming convex, and higher-order of smoothness of the objective function, this paper provides a zero-order accelerated stochastic gradient descent (ZO-AccSGD) method for solving this problem, which exploits the higher-order of smoothness information via kernel approximation. As theoretical results, we show that the ZO-AccSGD algorithm proposed in this paper improves the convergence results of state-of-the-art (SOTA) algorithms, namely the estimate of iteration complexity. In addition, our theoretical analysis provides an estimate of the maximum allowable noise level at which the desired accuracy can be achieved. Validation of our theoretical results is demonstrated both on the model function and on functions of interest in the field of machine learning. We also provide a discussion in which we explain the results obtained and the superiority of the proposed algorithm over SOTA algorithms for solving the original problem.

Suggested Citation

  • Aleksandr Lobanov & Nail Bashirov & Alexander Gasnikov, 2024. "The “Black-Box” Optimization Problem: Zero-Order Accelerated Stochastic Method via Kernel Approximation," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2451-2486, December.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:3:d:10.1007_s10957-024-02556-6
    DOI: 10.1007/s10957-024-02556-6
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    References listed on IDEAS

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    1. repec:inm:orijoo:v:5:y:2023:i:3:p:256-272 is not listed on IDEAS
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, April.
    3. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Katya Scheinberg, 2022. "Finite Difference Gradient Approximation: To Randomize or Not?," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2384-2388, September.
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