IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v84y2023i2d10.1007_s10589-022-00421-8.html
   My bibliography  Save this article

On FISTA with a relative error rule

Author

Listed:
  • Yunier Bello-Cruz

    (Northern Illinois University)

  • Max L. N. Gonçalves

    (Universidade Federal de Goiás)

  • Nathan Krislock

    (Northern Illinois University)

Abstract

The fast iterative shrinkage/thresholding algorithm (FISTA) is one of the most popular first-order iterations for minimizing the sum of two convex functions. FISTA is known to improve the complexity of the classical proximal gradient method (PGM) from $$O(k^{-1})$$ O ( k - 1 ) to the optimal complexity $$O(k^{-2})$$ O ( k - 2 ) in terms of the sequence of the functional values. When the evaluation of the proximal operator is hard, inexact versions of FISTA might be used to solve the problem. In this paper, we proposed an inexact version of FISTA by solving the proximal subproblem inexactly using a relative error criterion instead of exogenous and diminishing error rules. The introduced relative error rule in the FISTA iteration is related to the progress of the algorithm at each step and does not increase the computational burden per iteration. Moreover, the proposed algorithm recovers the same optimal convergence rate as FISTA. Some numerical experiments are also reported to illustrate the numerical behavior of the relative inexact method when compared with FISTA under an absolute error criterion.

Suggested Citation

  • Yunier Bello-Cruz & Max L. N. Gonçalves & Nathan Krislock, 2023. "On FISTA with a relative error rule," Computational Optimization and Applications, Springer, vol. 84(2), pages 295-318, March.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:2:d:10.1007_s10589-022-00421-8
    DOI: 10.1007/s10589-022-00421-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-022-00421-8
    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/s10589-022-00421-8?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. Lewandowski, Daniel & Kurowicka, Dorota & Joe, Harry, 2009. "Generating random correlation matrices based on vines and extended onion method," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 1989-2001, October.
    2. R. Díaz Millán & M. Pentón Machado, 2019. "Inexact proximal $$\epsilon $$ϵ-subgradient methods for composite convex optimization problems," Journal of Global Optimization, Springer, vol. 75(4), pages 1029-1060, December.
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Hedy Attouch & Alexandre Cabot & Zaki Chbani & Hassan Riahi, 2018. "Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 1-36, October.
    5. Jonathan Eckstein & Wang Yao, 2017. "Approximate ADMM algorithms derived from Lagrangian splitting," Computational Optimization and Applications, Springer, vol. 68(2), pages 363-405, November.
    6. Vando A. Adona & Max L. N. Gonçalves & Jefferson G. Melo, 2019. "A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 640-666, August.
    7. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    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. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2020. "An inexact proximal generalized alternating direction method of multipliers," Computational Optimization and Applications, Springer, vol. 76(3), pages 621-647, July.
    2. Mauricio Romero Sicre, 2020. "On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 991-1019, July.
    3. Kaiwen Ma & Nikolaos V. Sahinidis & Sreekanth Rajagopalan & Satyajith Amaran & Scott J Bury, 2021. "Decomposition in derivative-free optimization," Journal of Global Optimization, Springer, vol. 81(2), pages 269-292, October.
    4. Hao Wang & Hao Zeng & Jiashan Wang, 2022. "An extrapolated iteratively reweighted $$\ell _1$$ ℓ 1 method with complexity analysis," Computational Optimization and Applications, Springer, vol. 83(3), pages 967-997, December.
    5. A. Scagliotti & P. Colli Franzone, 2022. "A piecewise conservative method for unconstrained convex optimization," Computational Optimization and Applications, Springer, vol. 81(1), pages 251-288, January.
    6. Ren Jiang & Zhifeng Ji & Wuling Mo & Suhua Wang & Mingjun Zhang & Wei Yin & Zhen Wang & Yaping Lin & Xueke Wang & Umar Ashraf, 2022. "A Novel Method of Deep Learning for Shear Velocity Prediction in a Tight Sandstone Reservoir," Energies, MDPI, vol. 15(19), pages 1-20, September.
    7. Zhang, Yixiao & Yu, Cindy L. & Li, Haitao, 2022. "Nowcasting GDP Using Dynamic Factor Model with Unknown Number of Factors and Stochastic Volatility: A Bayesian Approach," Econometrics and Statistics, Elsevier, vol. 24(C), pages 75-93.
    8. Alejandro Plastina & Sergio H. Lence & Ariel Ortiz‐Bobea, 2021. "How weather affects the decomposition of total factor productivity in U.S. agriculture," Agricultural Economics, International Association of Agricultural Economists, vol. 52(2), pages 215-234, March.
    9. Matthias Breuer & Harm H. Schütt, 2023. "Accounting for uncertainty: an application of Bayesian methods to accruals models," Review of Accounting Studies, Springer, vol. 28(2), pages 726-768, June.
    10. William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    11. Flórez, Alvaro J. & Molenberghs, Geert & Van der Elst, Wim & Alonso Abad, Ariel, 2022. "An efficient algorithm to assess multivariate surrogate endpoints in a causal inference framework," Computational Statistics & Data Analysis, Elsevier, vol. 172(C).
    12. Zhaosong Lu & Xiaojun Chen, 2018. "Generalized Conjugate Gradient Methods for ℓ 1 Regularized Convex Quadratic Programming with Finite Convergence," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 275-303, February.
    13. Giuseppe Brandi & Ruggero Gramatica & Tiziana Di Matteo, 2019. "Unveil stock correlation via a new tensor-based decomposition method," Papers 1911.06126, arXiv.org, revised Apr 2020.
    14. Masaru Ito, 2016. "New results on subgradient methods for strongly convex optimization problems with a unified analysis," Computational Optimization and Applications, Springer, vol. 65(1), pages 127-172, September.
    15. Andrew Y. Chen & Jack McCoy, 2022. "Missing Values Handling for Machine Learning Portfolios," Papers 2207.13071, arXiv.org, revised Jan 2024.
    16. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    17. Gregory Benton & Wesley J. Maddox & Andrew Gordon Wilson, 2022. "Volatility Based Kernels and Moving Average Means for Accurate Forecasting with Gaussian Processes," Papers 2207.06544, arXiv.org.
    18. Juho Kettunen & Lauri Mehtätalo & Eeva‐Stiina Tuittila & Aino Korrensalo & Jarno Vanhatalo, 2024. "Joint species distribution modeling with competition for space," Environmetrics, John Wiley & Sons, Ltd., vol. 35(2), March.
    19. 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.
    20. Dimitris Bertsimas & Ryan Cory-Wright, 2022. "A Scalable Algorithm for Sparse Portfolio Selection," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1489-1511, May.

    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:coopap:v:84:y:2023:i:2:d:10.1007_s10589-022-00421-8. 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.