IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v75y2020i1d10.1007_s10589-019-00140-7.html
   My bibliography  Save this article

Markov chain block coordinate descent

Author

Listed:
  • Tao Sun

    (National University of Defense Technology)

  • Yuejiao Sun

    (University of California)

  • Yangyang Xu

    (Rensselaer Polytechnic Institute)

  • Wotao Yin

    (University of California)

Abstract

The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind of block selection is neither i.i.d. random nor cyclic. On the other hand, it is a natural choice for some applications in distributed optimization and Markov decision process, where i.i.d. random and cyclic selections are either infeasible or very expensive. By applying mixing-time properties of a Markov chain, we prove convergence of Markov chain BCD for minimizing Lipschitz differentiable functions, which can be nonconvex. When the functions are convex and strongly convex, we establish both sublinear and linear convergence rates, respectively. We also present a method of Markov chain inertial BCD. Finally, we discuss potential applications.

Suggested Citation

  • Tao Sun & Yuejiao Sun & Yangyang Xu & Wotao Yin, 2020. "Markov chain block coordinate descent," Computational Optimization and Applications, Springer, vol. 75(1), pages 35-61, January.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:1:d:10.1007_s10589-019-00140-7
    DOI: 10.1007/s10589-019-00140-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-019-00140-7
    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/s10589-019-00140-7?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. 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. Brucker, Peter & Drexl, Andreas & Mohring, Rolf & Neumann, Klaus & Pesch, Erwin, 1999. "Resource-constrained project scheduling: Notation, classification, models, and methods," European Journal of Operational Research, Elsevier, vol. 112(1), pages 3-41, January.
    3. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. Sjur Didrik Flåm, 2019. "Blocks of coordinates, stochastic programming, and markets," Computational Management Science, Springer, vol. 16(1), pages 3-16, February.
    3. 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.
    4. 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.
    5. 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.
    6. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    7. Jin Zhang & Xide Zhu, 2022. "Linear Convergence of Prox-SVRG Method for Separable Non-smooth Convex Optimization Problems under Bounded Metric Subregularity," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 564-597, February.
    8. Zhigang Li & Mingchuan Zhang & Junlong Zhu & Ruijuan Zheng & Qikun Zhang & Qingtao Wu, 2018. "Stochastic Block-Coordinate Gradient Projection Algorithms for Submodular Maximization," Complexity, Hindawi, vol. 2018, pages 1-11, December.
    9. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    10. A. Ghaffari-Hadigheh & L. Sinjorgo & R. Sotirov, 2024. "On convergence of a q-random coordinate constrained algorithm for non-convex problems," Journal of Global Optimization, Springer, vol. 90(4), pages 843-868, December.
    11. Ching-pei Lee & Stephen J. Wright, 2020. "Inexact Variable Metric Stochastic Block-Coordinate Descent for Regularized Optimization," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 151-187, April.
    12. R. Lopes & S. A. Santos & P. J. S. Silva, 2019. "Accelerating block coordinate descent methods with identification strategies," Computational Optimization and Applications, Springer, vol. 72(3), pages 609-640, April.
    13. Yangyang Xu, 2019. "Asynchronous parallel primal–dual block coordinate update methods for affinely constrained convex programs," Computational Optimization and Applications, Springer, vol. 72(1), pages 87-113, January.
    14. Mingyi Hong & Tsung-Hui Chang & Xiangfeng Wang & Meisam Razaviyayn & Shiqian Ma & Zhi-Quan Luo, 2020. "A Block Successive Upper-Bound Minimization Method of Multipliers for Linearly Constrained Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 833-861, August.
    15. Cassioli, A. & Di Lorenzo, D. & Sciandrone, M., 2013. "On the convergence of inexact block coordinate descent methods for constrained optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 274-281.
    16. Rachael Tappenden & Peter Richtárik & Jacek Gondzio, 2016. "Inexact Coordinate Descent: Complexity and Preconditioning," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 144-176, July.
    17. Asbach, Lasse & Dorndorf, Ulrich & Pesch, Erwin, 2009. "Analysis, modeling and solution of the concrete delivery problem," European Journal of Operational Research, Elsevier, vol. 193(3), pages 820-835, March.
    18. Wendi Tian & Erik Demeulemeester, 2014. "Railway scheduling reduces the expected project makespan over roadrunner scheduling in a multi-mode project scheduling environment," Annals of Operations Research, Springer, vol. 213(1), pages 271-291, February.
    19. Byung-Cheon Choi & Changmuk Kang, 2019. "A linear time–cost tradeoff problem with multiple milestones under a comb graph," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 341-361, August.
    20. Min Tao & Jiang-Ning Li, 2023. "Error Bound and Isocost Imply Linear Convergence of DCA-Based Algorithms to D-Stationarity," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 205-232, April.

    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:75:y:2020:i:1:d:10.1007_s10589-019-00140-7. 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.