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General Hölder Smooth Convergence Rates Follow from Specialized Rates Assuming Growth Bounds

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  • Benjamin Grimmer

    (Johns Hopkins University)

Abstract

Often in the analysis of first-order methods for both smooth and nonsmooth optimization, assuming the existence of a growth/error bound or KL condition facilitates much stronger convergence analysis. Hence, separate analysis is typically needed for the general case and for the growth bounded cases. We give meta-theorems for deriving general convergence rates from those assuming a growth lower bound. Applying this simple but conceptually powerful tool to the proximal point, subgradient, bundle, dual averaging, gradient descent, Frank–Wolfe and universal accelerated methods immediately recovers their known convergence rates for general convex optimization problems from their specialized rates. New convergence results follow for bundle methods, dual averaging and Frank–Wolfe. Our results can lift any rate based on Hölder continuous gradients and Hölder growth bounds. Moreover, our theory provides simple proofs of optimal convergence lower bounds under Hölder growth from textbook examples without growth bounds.

Suggested Citation

  • Benjamin Grimmer, 2023. "General Hölder Smooth Convergence Rates Follow from Specialized Rates Assuming Growth Bounds," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 51-70, April.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:1:d:10.1007_s10957-023-02178-4
    DOI: 10.1007/s10957-023-02178-4
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    References listed on IDEAS

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    1. Yu Du & Andrzej Ruszczyński, 2017. "Rate of Convergence of the Bundle Method," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 908-922, June.
    2. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Marguerite Frank & Philip Wolfe, 1956. "An algorithm for quadratic programming," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 95-110, March.
    4. 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).
    5. K. C. Kiwiel, 2000. "Efficiency of Proximal Bundle Methods," Journal of Optimization Theory and Applications, Springer, vol. 104(3), pages 589-603, March.
    6. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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