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Stochastic Intermediate Gradient Method for Convex Problems with Stochastic Inexact Oracle

Author

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  • Pavel Dvurechensky

    (Weierstrass Institute for Applied Analysis and Stochastics
    Institute for Information Transmission Problems RAS)

  • Alexander Gasnikov

    (Institute for Information Transmission Problems RAS
    Moscow Institute of Physics and Technology)

Abstract

In this paper, we introduce new methods for convex optimization problems with stochastic inexact oracle. Our first method is an extension of the Intermediate Gradient Method proposed by Devolder, Glineur and Nesterov for problems with deterministic inexact oracle. Our method can be applied to problems with composite objective function, both deterministic and stochastic inexactness of the oracle, and allows using a non-Euclidean setup. We estimate the rate of convergence in terms of the expectation of the non-optimality gap and provide a way to control the probability of large deviations from this rate. Also we introduce two modifications of this method for strongly convex problems. For the first modification, we estimate the rate of convergence for the non-optimality gap expectation and, for the second, we provide a bound for the probability of large deviations from the rate of convergence in terms of the expectation of the non-optimality gap. All the rates lead to the complexity estimates for the proposed methods, which up to a multiplicative constant coincide with the lower complexity bound for the considered class of convex composite optimization problems with stochastic inexact oracle.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0999-6
    DOI: 10.1007/s10957-016-0999-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2013. "Intermediate gradient methods for smooth convex problems with inexact oracle," LIDAM Discussion Papers CORE 2013017, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV, Yu., 2012. "Subgradient methods for huge-scale optimization problems," LIDAM Discussion Papers CORE 2012002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," LIDAM Discussion Papers CORE 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Olivier Fercoq & Zheng Qu, 2020. "Restarting the accelerated coordinate descent method with a rough strong convexity estimate," Computational Optimization and Applications, Springer, vol. 75(1), pages 63-91, January.
    2. 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.
    3. 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.
    4. Vladimir Krutikov & Svetlana Gutova & Elena Tovbis & Lev Kazakovtsev & Eugene Semenkin, 2022. "Relaxation Subgradient Algorithms with Machine Learning Procedures," Mathematics, MDPI, vol. 10(21), pages 1-33, October.
    5. Tianxiao Sun & Ion Necoara & Quoc Tran-Dinh, 2020. "Composite convex optimization with global and local inexact oracles," Computational Optimization and Applications, Springer, vol. 76(1), pages 69-124, May.

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