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Inexact Variable Metric Stochastic Block-Coordinate Descent for Regularized Optimization

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Listed:
  • Ching-pei Lee

    (National University of Singapore)

  • Stephen J. Wright

    (University of Wisconsin-Madison)

Abstract

Block-coordinate descent is a popular framework for large-scale regularized optimization problems with block-separable structure. Existing methods have several limitations. They often assume that subproblems can be solved exactly at each iteration, which in practical terms usually restricts the quadratic term in the subproblem to be diagonal, thus losing most of the benefits of higher-order derivative information. Moreover, in contrast to the smooth case, non-uniform sampling of the blocks has not yet been shown to improve the convergence rate bounds for regularized problems. This work proposes an inexact randomized block-coordinate descent method based on a regularized quadratic subproblem, in which the quadratic term can vary from iteration to iteration: a “variable metric.” We provide a detailed convergence analysis for both convex and non-convex problems. Our analysis generalizes, to the regularized case, Nesterov’s proposal for improving convergence of block-coordinate descent by sampling proportional to the blockwise Lipschitz constants. We improve the convergence rate in the convex case by weakening the dependency on the initial objective value. Empirical results also show that significant benefits accrue from the use of a variable metric.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:185:y:2020:i:1:d:10.1007_s10957-020-01639-4
    DOI: 10.1007/s10957-020-01639-4
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    References listed on IDEAS

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    1. 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.
    2. Hiva Ghanbari & Katya Scheinberg, 2018. "Proximal quasi-Newton methods for regularized convex optimization with linear and accelerated sublinear convergence rates," Computational Optimization and Applications, Springer, vol. 69(3), pages 597-627, April.
    3. 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.
    4. Ching-pei Lee & Stephen J. Wright, 2019. "Inexact Successive quadratic approximation for regularized optimization," Computational Optimization and Applications, Springer, vol. 72(3), pages 641-674, 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. Lukas Meier & Sara Van De Geer & Peter Bühlmann, 2008. "The group lasso for logistic regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 53-71, February.
    7. 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).
    8. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    9. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2016. "A block coordinate variable metric forward–backward algorithm," Journal of Global Optimization, Springer, vol. 66(3), pages 457-485, November.
    10. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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