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Accelerated Bregman proximal gradient methods for relatively smooth convex optimization

Author

Listed:
  • Filip Hanzely

    (King Abdullah University of Science and Technology (KAUST)
    Toyota Technological Institute at Chicago (TTIC))

  • Peter Richtárik

    (King Abdullah University of Science and Technology (KAUST)
    Moscow Institute of Physics and Technology)

  • Lin Xiao

    (Microsoft Research)

Abstract

We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. We investigate a triangle scaling property of the Bregman distance generated by the reference convex function and present accelerated Bregman proximal gradient (ABPG) methods that attain an $$O(k^{-\gamma })$$ O ( k - γ ) convergence rate, where $$\gamma \in (0,2]$$ γ ∈ ( 0 , 2 ] is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have $$\gamma =2$$ γ = 2 and recover the convergence rate of Nesterov’s accelerated gradient methods. For non-Euclidean Bregman distances, the TSE can be much smaller (say $$\gamma \le 1$$ γ ≤ 1 ), but we show that a relaxed definition of intrinsic TSE is always equal to 2. We exploit the intrinsic TSE to develop adaptive ABPG methods that converge much faster in practice. Although theoretical guarantees on a fast convergence rate seem to be out of reach in general, our methods obtain empirical $$O(k^{-2})$$ O ( k - 2 ) rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.

Suggested Citation

  • Filip Hanzely & Peter Richtárik & Lin Xiao, 2021. "Accelerated Bregman proximal gradient methods for relatively smooth convex optimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 405-440, June.
  • Handle: RePEc:spr:coopap:v:79:y:2021:i:2:d:10.1007_s10589-021-00273-8
    DOI: 10.1007/s10589-021-00273-8
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    5. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    6. Yi Zhou & Yingbin Liang & Lixin Shen, 2019. "A simple convergence analysis of Bregman proximal gradient algorithm," Computational Optimization and Applications, Springer, vol. 73(3), pages 903-912, July.
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    Cited by:

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    2. Yin Liu & Sam Davanloo Tajbakhsh, 2023. "Stochastic Composition Optimization of Functions Without Lipschitz Continuous Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 239-289, July.

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