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FISTA Restart Using an Automatic Estimation of the Growth Parameter

Author

Listed:
  • Jean-François Aujol

    (Univ. Bordeaux, Bordeaux INP, CNRS, IMB)

  • Charles Dossal

    (IMT, Univ. Toulouse, INSA Toulouse)

  • Hippolyte Labarrière

    (MaLGa, DIBRIS, Università degli Studi di Genova)

  • Aude Rondepierre

    (IMT, Univ. Toulouse, INSA Toulouse
    LAAS, Univ. Toulouse, CNRS)

Abstract

In this paper, we propose a restart scheme for FISTA (Fast Iterative Shrinking-Threshold Algorithm) [6]. This method which is a generalisation of Nesterov’s accelerated gradient algorithm [23] is widely used in the field of large-scale convex optimization problems as it ensures a quadratic decrease $$o\left( 1/k^2\right) $$ o 1 / k 2 of the error for convex functions [3, 12]. When considering a function that satisfies stronger assumptions such as strong convexity or quadratic growth, several methods provide faster convergence rates by taking advantage of this geometry property, including FISTA restart schemes. In particular, the schemes that provide the fastest theoretical convergence rates rely on the growth parameter of the function, which is generally difficult to estimate. Recent works [1, 2] show that restarting FISTA can ensure a fast convergence for functions having a quadratic growth without requiring any knowledge on the growth parameter. We improve these restart schemes by providing a better asymptotical convergence rate and by requiring a lower computational cost. We illustrate our theoretical results with some numerical examples.

Suggested Citation

  • Jean-François Aujol & Charles Dossal & Hippolyte Labarrière & Aude Rondepierre, 2025. "FISTA Restart Using an Automatic Estimation of the Growth Parameter," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-27, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02688-3
    DOI: 10.1007/s10957-025-02688-3
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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).
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