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Exact worst-case convergence rates of the proximal gradient method for composite convex minimization

Author

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  • Adrien B. Taylor
  • Julien M. Hendrickx
  • François Glineur

Abstract

We study the worst-case convergence rates of the proximal gradient method for minimizing the sum of a smooth strongly convex function and a non-smooth convex function, whose proximal operator is available. We establish the exact worst-case convergence rates of the proximal gradient method in this setting for any step size and for different standard performance measures: objective function accuracy, distance to optimality and residual gradient norm. The proof methodology relies on recent developments in performance estimation of first-order methods, based on semidefinite programming. In the case of the proximal gradient method, this methodology allows obtaining exact and non-asymptotic worst-case guarantees that are conceptually very simple, although apparently new. On the way, we discuss how strong convexity can be replaced by weaker assumptions, while preserving the corresponding convergence rates. We also establish that the same fixed step size policy is optimal for all three performance measures. Finally, we extend recent results on the worst-case behavior of gradient descent with exact line search to the proximal case.
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Suggested Citation

  • Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact worst-case convergence rates of the proximal gradient method for composite convex minimization," LIDAM Reprints CORE 2975, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:2975
    Note: In : Journal of Optimization Theory and Applications, 178, 455-476, 2018
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    Cited by:

    1. Zaineb Amor & Philippe Ciuciu & Chaithya G. R. & Guillaume Daval-Frérot & Franck Mauconduit & Bertrand Thirion & Alexandre Vignaud, 2024. "Non-Cartesian 3D-SPARKLING vs Cartesian 3D-EPI encoding schemes for functional Magnetic Resonance Imaging at 7 Tesla," PLOS ONE, Public Library of Science, vol. 19(5), pages 1-30, May.
    2. André Uschmajew & Bart Vandereycken, 2022. "A Note on the Optimal Convergence Rate of Descent Methods with Fixed Step Sizes for Smooth Strongly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 364-373, July.
    3. Abbaszadehpeivasti, Hadi, 2024. "Performance analysis of optimization methods for machine learning," Other publications TiSEM 3050a62d-1a1f-494e-99ef-7, Tilburg University, School of Economics and Management.
    4. Sandra S. Y. Tan & Antonios Varvitsiotis & Vincent Y. F. Tan, 2021. "Analysis of Optimization Algorithms via Sum-of-Squares," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 56-81, July.
    5. Wei Peng & Hui Zhang & Xiaoya Zhang & Lizhi Cheng, 2020. "Global complexity analysis of inexact successive quadratic approximation methods for regularized optimization under mild assumptions," Journal of Global Optimization, Springer, vol. 78(1), pages 69-89, September.
    6. Guoyong Gu & Junfeng Yang, 2024. "Tight Ergodic Sublinear Convergence Rate of the Relaxed Proximal Point Algorithm for Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 373-387, July.
    7. Donghwan Kim & Jeffrey A. Fessler, 2021. "Optimizing the Efficiency of First-Order Methods for Decreasing the Gradient of Smooth Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 192-219, January.

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