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An accelerated directional derivative method for smooth stochastic convex optimization

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  • Dvurechensky, Pavel
  • Gorbunov, Eduard
  • Gasnikov, Alexander

Abstract

We consider smooth stochastic convex optimization problems in the context of algorithms which are based on directional derivatives of the objective function. This context can be considered as an intermediate one between derivative-free optimization and gradient-based optimization. We assume that at any given point and for any given direction, a stochastic approximation for the directional derivative of the objective function at this point and in this direction is available with some additive noise. The noise is assumed to be of an unknown nature, but bounded in the absolute value. We underline that we consider directional derivatives in any direction, as opposed to coordinate descent methods which use only derivatives in coordinate directions. For this setting, we propose a non-accelerated and an accelerated directional derivative method and provide their complexity bounds. Our non-accelerated algorithm has a complexity bound which is similar to the gradient-based algorithm, that is, without any dimension-dependent factor. Our accelerated algorithm has a complexity bound which coincides with the complexity bound of the accelerated gradient-based algorithm up to a factor of square root of the problem dimension. We extend these results to strongly convex problems.

Suggested Citation

  • Dvurechensky, Pavel & Gorbunov, Eduard & Gasnikov, Alexander, 2021. "An accelerated directional derivative method for smooth stochastic convex optimization," European Journal of Operational Research, Elsevier, vol. 290(2), pages 601-621.
    • Handle: RePEc:eee:ejores:v:290:y:2021:i:2:p:601-621
    DOI: 10.1016/j.ejor.2020.08.027
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    References listed on IDEAS

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    1. 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).
    2. Yurii NESTEROV & Sebastian U. STICH, 2017. "Efficiency of the accelerated coordinate descent method on structured optimization problems," LIDAM Reprints CORE 2845, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. DEVOLDER, Olivier, 2011. "Stochastic first order methods in smooth convex optimization," LIDAM Discussion Papers CORE 2011070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Powell, Warren B., 2019. "A unified framework for stochastic optimization," European Journal of Operational Research, Elsevier, vol. 275(3), pages 795-821.
    5. Pavel Dvurechensky & Alexander Gasnikov, 2016. "Stochastic Intermediate Gradient Method for Convex Problems with Stochastic Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 121-145, October.
    6. 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).
    7. 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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    Cited by:

    1. Fedor Stonyakin & Alexander Gasnikov & Pavel Dvurechensky & Alexander Titov & Mohammad Alkousa, 2022. "Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 988-1013, September.
    2. Anastasiya Ivanova & Pavel Dvurechensky & Evgeniya Vorontsova & Dmitry Pasechnyuk & Alexander Gasnikov & Darina Dvinskikh & Alexander Tyurin, 2022. "Oracle Complexity Separation in Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 462-490, June.

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