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Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact

Author

Listed:
  • Nikita Kornilov

    (Moscow Institute of Physics and Technology)

  • Alexander Gasnikov

    (Moscow Institute of Physics and Technology
    Skoltech
    ISP RAS Research Center for Trusted Artificial Intelligence)

  • Pavel Dvurechensky

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • Darina Dvinskikh

    (HSE University)

Abstract

We present two easy-to-implement gradient-free/zeroth-order methods to optimize a stochastic non-smooth function accessible only via a black-box. The methods are built upon efficient first-order methods in the heavy-tailed case, i.e., when the gradient noise has infinite variance but bounded $$(1+\kappa)$$ ( 1 + κ ) -th moment for some $$\kappa \in(0,1]$$ κ ∈ ( 0 , 1 ] . The first algorithm is based on the stochastic mirror descent with a particular class of uniformly convex mirror maps which is robust to heavy-tailed noise. The second algorithm is based on the stochastic mirror descent and gradient clipping technique. Additionally, for the objective functions satisfying the r-growth condition, faster algorithms are proposed based on these methods and the restart technique.

Suggested Citation

  • Nikita Kornilov & Alexander Gasnikov & Pavel Dvurechensky & Darina Dvinskikh, 2023. "Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact," Computational Management Science, Springer, vol. 20(1), pages 1-43, December.
    • Handle: RePEc:spr:comgts:v:20:y:2023:i:1:d:10.1007_s10287-023-00470-2
    DOI: 10.1007/s10287-023-00470-2
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    References listed on IDEAS

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    1. 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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