IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v36y2024i6p1611-1633.html
   My bibliography  Save this article

Convergence Rates of Zeroth Order Gradient Descent for Łojasiewicz Functions

Author

Listed:
  • Tianyu Wang

    (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200437, China; Shanghai Artificial Intelligence Laboratory, Shanghai 200232, China)

  • Yasong Feng

    (Shanghai Center for Mathematical Sciences, Fudan University, Shanghai 200437, China)

Abstract

We prove convergence rates of Zeroth-order Gradient Descent (ZGD) algorithms for Łojasiewicz functions. Our results show that for smooth Łojasiewicz functions with Łojasiewicz exponent larger than 0.5 and smaller than 1, the functions values can converge much faster than the (zeroth-order) gradient descent trajectory. Similar results hold for convex nonsmooth Łojasiewicz functions.

Suggested Citation

  • Tianyu Wang & Yasong Feng, 2024. "Convergence Rates of Zeroth Order Gradient Descent for Łojasiewicz Functions," INFORMS Journal on Computing, INFORMS, vol. 36(6), pages 1611-1633, December.
    • Handle: RePEc:inm:orijoc:v:36:y:2024:i:6:p:1611-1633
    DOI: 10.1287/ijoc.2023.0247
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/ijoc.2023.0247
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2023.0247?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
    ---><---

    References listed on IDEAS

    as
    1. 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).
    2. Yurii NESTEROV & Vladimir SPOKOINY, 2017. "Random gradient-free minimization of convex functions," LIDAM Reprints CORE 2851, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    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. 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.
    2. 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).
    3. 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.
    4. Chuan He & Heng Huang & Zhaosong Lu, 2024. "A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization," Computational Optimization and Applications, Springer, vol. 89(3), pages 843-894, December.
    5. V. Kungurtsev & F. Rinaldi, 2021. "A zeroth order method for stochastic weakly convex optimization," Computational Optimization and Applications, Springer, vol. 80(3), pages 731-753, December.
    6. Hoang Tran & Qiang Du & Guannan Zhang, 2025. "Convergence analysis for a nonlocal gradient descent method via directional Gaussian smoothing," Computational Optimization and Applications, Springer, vol. 90(2), pages 481-513, March.
    7. Kenji Ueda & Nobuo Yamashita, 2012. "Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 450-467, February.
    8. Yamakawa, Yuya & Yamashita, Nobuo, 2025. "Convergence analysis of a regularized Newton method with generalized regularization terms for unconstrained convex optimization problems," Applied Mathematics and Computation, Elsevier, vol. 491(C).
    9. Jean-Jacques Forneron, 2023. "Noisy, Non-Smooth, Non-Convex Estimation of Moment Condition Models," Papers 2301.07196, arXiv.org, revised Aug 2025.
    10. Liaoyuan Zeng & Ting Kei Pong, 2022. "$$\rho$$ ρ -regularization subproblems: strong duality and an eigensolver-based algorithm," Computational Optimization and Applications, Springer, vol. 81(2), pages 337-368, March.
    11. Alireza Aghasi & Saeed Ghadimi, 2025. "Fully Zeroth-Order Bilevel Programming via Gaussian Smoothing," Journal of Optimization Theory and Applications, Springer, vol. 205(2), pages 1-39, May.
    12. Fedor Stonyakin & Ilya Kuruzov & Boris Polyak, 2023. "Stopping Rules for Gradient Methods for Non-convex Problems with Additive Noise in Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 531-551, August.
    13. Yuquan Chen & Yunkang Sun & Bing Wang, 2023. "Improving the Performance of Optimization Algorithms Using the Adaptive Fixed-Time Scheme and Reset Scheme," Mathematics, MDPI, vol. 11(22), pages 1-16, November.
    14. Hong Zhu & Yunhai Xiao, 2024. "A hybrid inexact regularized Newton and negative curvature method," Computational Optimization and Applications, Springer, vol. 88(3), pages 849-870, July.
    15. 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.
    16. Aleksandr Lobanov & Andrew Veprikov & Georgiy Konin & Aleksandr Beznosikov & Alexander Gasnikov & Dmitry Kovalev, 2023. "Non-smooth setting of stochastic decentralized convex optimization problem over time-varying Graphs," Computational Management Science, Springer, vol. 20(1), pages 1-55, December.
    17. Jaroslav Fowkes & Nicholas Gould & Chris Farmer, 2013. "A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions," Journal of Global Optimization, Springer, vol. 56(4), pages 1791-1815, August.
    18. Stefan Wager & Kuang Xu, 2021. "Experimenting in Equilibrium," Management Science, INFORMS, vol. 67(11), pages 6694-6715, November.
    19. Nicholas I. M. Gould & Tyrone Rees & Jennifer A. Scott, 2019. "Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 1-35, May.
    20. Nadav Hallak & Marc Teboulle, 2020. "Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 480-503, August.

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:inm:orijoc:v:36:y:2024:i:6:p:1611-1633. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.