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A flexible coordinate descent method

Author

Listed:
  • Kimon Fountoulakis

    (University of California Berkeley)

  • Rachael Tappenden

    (University of Canterbury)

Abstract

We present a novel randomized block coordinate descent method for the minimization of a convex composite objective function. The method uses (approximate) partial second-order (curvature) information, so that the algorithm performance is more robust when applied to highly nonseparable or ill conditioned problems. We call the method Flexible Coordinate Descent (FCD). At each iteration of FCD, a block of coordinates is sampled randomly, a quadratic model is formed about that block and the model is minimized approximately/inexactly to determine the search direction. An inexpensive line search is then employed to ensure a monotonic decrease in the objective function and acceptance of large step sizes. We present several high probability iteration complexity results to show that convergence of FCD is guaranteed theoretically. Finally, we present numerical results on large-scale problems to demonstrate the practical performance of the method.

Suggested Citation

  • Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:2:d:10.1007_s10589-018-9984-3
    DOI: 10.1007/s10589-018-9984-3
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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. Kimon Fountoulakis & Jacek Gondzio, 2016. "Performance of first- and second-order methods for $$\ell _1$$ ℓ 1 -regularized least squares problems," Computational Optimization and Applications, Springer, vol. 65(3), pages 605-635, December.
    3. 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).
    4. Paul Tseng & Sangwoon Yun, 2010. "A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training," Computational Optimization and Applications, Springer, vol. 47(2), pages 179-206, October.
    5. 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).
    6. 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.
    7. 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.
    8. 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.
    9. 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.
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    Cited by:

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    2. 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.
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    4. 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.
    5. Bastian Pötzl & Anton Schiela & Patrick Jaap, 2022. "Second order semi-smooth Proximal Newton methods in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 82(2), pages 465-498, June.

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