IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v191y2021i1d10.1007_s10957-021-01930-y.html
   My bibliography  Save this article

Superfast Second-Order Methods for Unconstrained Convex Optimization

Author

Listed:
  • Yurii Nesterov

    (Catholic University of Louvain (UCL))

Abstract

In this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is $$O\left( k^{-7/2} \right) $$ O k - 7 / 2 . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed $$O\left( \ln {1 \over \epsilon }\right) $$ O ln 1 ϵ times, where $$\epsilon $$ ϵ is the desired accuracy of the solution for our problem.

Suggested Citation

  • Yurii Nesterov, 2021. "Superfast Second-Order Methods for Unconstrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 1-30, October.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01930-y
    DOI: 10.1007/s10957-021-01930-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01930-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-021-01930-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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,, 2019. "Inexact basic tensor methods," LIDAM Discussion Papers CORE 2019023, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, June.
    4. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Nesterov, Yurii, 2022. "Quartic Regularity," LIDAM Discussion Papers CORE 2022001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Xin Jiang & Lieven Vandenberghe, 2023. "Bregman Three-Operator Splitting Methods," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 936-972, March.
    2. Nesterov, Yurii, 2022. "Quartic Regularity," LIDAM Discussion Papers CORE 2022001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Doikov, Nikita & Nesterov, Yurii, 2020. "Affine-invariant contracting-point methods for Convex Optimization," LIDAM Discussion Papers CORE 2020029, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
    5. Emanuel Laude & Peter Ochs & Daniel Cremers, 2020. "Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 724-761, March.
    6. 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.
    7. Radu-Alexandru Dragomir & Alexandre d’Aspremont & Jérôme Bolte, 2021. "Quartic First-Order Methods for Low-Rank Minimization," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 341-363, May.
    8. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    9. Alberto De Marchi & Andreas Themelis, 2022. "Proximal Gradient Algorithms Under Local Lipschitz Gradient Continuity," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 771-794, September.
    10. Masoud Ahookhosh, 2019. "Accelerated first-order methods for large-scale convex optimization: nearly optimal complexity under strong convexity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 319-353, June.
    11. Mahesh Chandra Mukkamala & Jalal Fadili & Peter Ochs, 2022. "Global convergence of model function based Bregman proximal minimization algorithms," Journal of Global Optimization, Springer, vol. 83(4), pages 753-781, August.
    12. Vincenzo Bonifaci, 2021. "A Laplacian approach to $$\ell _1$$ ℓ 1 -norm minimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 441-469, June.
    13. Chung-En Tsai & Hao-Chung Cheng & Yen-Huan Li, 2022. "Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States," Papers 2210.00997, arXiv.org, revised Sep 2023.
    14. Hui Zhang & Yu-Hong Dai & Lei Guo & Wei Peng, 2021. "Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 61-81, February.
    15. Zhongming Wu & Chongshou Li & Min Li & Andrew Lim, 2021. "Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 79(3), pages 617-644, March.
    16. 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.
    17. Shota Takahashi & Mituhiro Fukuda & Mirai Tanaka, 2022. "New Bregman proximal type algorithms for solving DC optimization problems," Computational Optimization and Applications, Springer, vol. 83(3), pages 893-931, December.
    18. Silvia Berra & Alessandro Torraca & Federico Benvenuto & Sara Sommariva, 2024. "Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 404-427, January.
    19. Ariizumi, Shumpei & Yamakawa, Yuya & Yamashita, Nobuo, 2024. "Convergence properties of Levenberg–Marquardt methods with generalized regularization terms," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    20. Seonho Park & Seung Hyun Jung & Panos M. Pardalos, 2020. "Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 953-971, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01930-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.