IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v77y2020i3d10.1007_s10589-020-00220-z.html
   My bibliography  Save this article

Momentum and stochastic momentum for stochastic gradient, Newton, proximal point and subspace descent methods

Author

Listed:
  • Nicolas Loizou

    (Université de Montréal)

  • Peter Richtárik

    (King Abdullah University of Science and Technology (KAUST))

Abstract

In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent: convex quadratic problems. We prove global non-asymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates, and dual function values. We also show that the primal iterates converge at an accelerated linear rate in a somewhat weaker sense. This is the first time a linear rate is shown for the stochastic heavy ball method (i.e., stochastic gradient descent method with momentum). Under somewhat weaker conditions, we establish a sublinear convergence rate for Cesàro averages of primal iterates. Moreover, we propose a novel concept, which we call stochastic momentum, aimed at decreasing the cost of performing the momentum step. We prove linear convergence of several stochastic methods with stochastic momentum, and show that in some sparse data regimes and for sufficiently small momentum parameters, these methods enjoy better overall complexity than methods with deterministic momentum. Finally, we perform extensive numerical testing on artificial and real datasets, including data coming from average consensus problems.

Suggested Citation

  • Nicolas Loizou & Peter Richtárik, 2020. "Momentum and stochastic momentum for stochastic gradient, Newton, proximal point and subspace descent methods," Computational Optimization and Applications, Springer, vol. 77(3), pages 653-710, December.
  • Handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00220-z
    DOI: 10.1007/s10589-020-00220-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00220-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-020-00220-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Gadat, Sébastien & Panloup, Fabien & Saadane, Sofiane, 2016. "Stochastic Heavy Ball," TSE Working Papers 16-712, Toulouse School of Economics (TSE).
    2. D. Leventhal & A. S. Lewis, 2010. "Randomized Methods for Linear Constraints: Convergence Rates and Conditioning," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 641-654, August.
    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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Emilie Chouzenoux & Jean-Baptiste Fest, 2022. "SABRINA: A Stochastic Subspace Majorization-Minimization Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 919-952, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Qin Wang & Weiguo Li & Wendi Bao & Feiyu Zhang, 2022. "Accelerated Randomized Coordinate Descent for Solving Linear Systems," Mathematics, MDPI, vol. 10(22), pages 1-20, November.
    2. Ruoyu Sun & Zhi-Quan Luo & Yinyu Ye, 2020. "On the Efficiency of Random Permutation for ADMM and Coordinate Descent," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 233-271, February.
    3. Du, Kui, 2024. "Regularized randomized iterative algorithms for factorized linear systems," Applied Mathematics and Computation, Elsevier, vol. 466(C).
    4. Xuefei Lu & Alessandro Rudi & Emanuele Borgonovo & Lorenzo Rosasco, 2020. "Faster Kriging: Facing High-Dimensional Simulators," Operations Research, INFORMS, vol. 68(1), pages 233-249, January.
    5. Costa, Manon & Gadat, Sébastien & Bercu, Bernard, 2020. "Stochastic approximation algorithms for superquantiles estimation," TSE Working Papers 20-1142, Toulouse School of Economics (TSE).
    6. Chen, Jia-Qi & Huang, Zheng-Da, 2020. "On the error estimate of the randomized double block Kaczmarz method," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    7. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Andrej Čopar & Blaž Zupan & Marinka Zitnik, 2019. "Fast optimization of non-negative matrix tri-factorization," PLOS ONE, Public Library of Science, vol. 14(6), pages 1-15, June.
    9. Duy Khuong Nguyen & Tu Bao Ho, 2017. "Accelerated parallel and distributed algorithm using limited internal memory for nonnegative matrix factorization," Journal of Global Optimization, Springer, vol. 68(2), pages 307-328, June.
    10. Abbaszadehpeivasti, Hadi & de Klerk, Etienne & Zamani, Moslem, 2023. "Convergence rate analysis of randomized and cyclic coordinate descent for convex optimization through semidefinite programming," Other publications TiSEM 88512ac0-c26a-4a99-b840-3, Tilburg University, School of Economics and Management.
    11. Ion Necoara & Andrei Patrascu, 2014. "A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints," Computational Optimization and Applications, Springer, vol. 57(2), pages 307-337, March.
    12. Sjur Didrik Flåm, 2019. "Blocks of coordinates, stochastic programming, and markets," Computational Management Science, Springer, vol. 16(1), pages 3-16, February.
    13. David Degras, 2021. "Sparse group fused lasso for model segmentation: a hybrid approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 625-671, September.
    14. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    15. Ion Necoara & Yurii Nesterov & François Glineur, 2017. "Random Block Coordinate Descent Methods for Linearly Constrained Optimization over Networks," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 227-254, April.
    16. Sjur Didrik Flåm, 2020. "Emergence of price-taking Behavior," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(3), pages 847-870, October.
    17. Fu, Sheng & Zhang, Sanguo & Liu, Yufeng, 2018. "Adaptively weighted large-margin angle-based classifiers," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 282-299.
    18. Andrei Patrascu & Ion Necoara, 2015. "Efficient random coordinate descent algorithms for large-scale structured nonconvex optimization," Journal of Global Optimization, Springer, vol. 61(1), pages 19-46, January.
    19. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    20. Chenxi Chen & Yunmei Chen & Yuyuan Ouyang & Eduardo Pasiliao, 2018. "Stochastic Accelerated Alternating Direction Method of Multipliers with Importance Sampling," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 676-695, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00220-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.