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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2407.17063 |
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| _version_ | 1866929435032158208 |
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| author | Aujol, Jean-François Dossal, Charles Labarrière, Hippolyte Rondepierre, Aude |
| author_facet | Aujol, Jean-François Dossal, Charles Labarrière, Hippolyte Rondepierre, Aude |
| contents | Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of $\mathcal{O}\left(n^{-2}\right)$ in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H\''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Strong Convergence of FISTA Iterates under H{ö}lderian and Quadratic Growth Conditions Aujol, Jean-François Dossal, Charles Labarrière, Hippolyte Rondepierre, Aude Optimization and Control Introduced by Beck and Teboulle, FISTA (for Fast Iterative Shrinkage-Thresholding Algorithm) is a first-order method widely used in convex optimization. Adapted from Nesterov's accelerated gradient method for convex functions, the generated sequence guarantees a decay of the function values of $\mathcal{O}\left(n^{-2}\right)$ in the convex setting. We show that for coercive functions satisfying some local growth condition (namely a H\''olderian or quadratic growth condition), this sequence strongly converges to a minimizer. This property, which has never been proved without assuming the uniqueness of the minimizer, is associated with improved convergence rates for the function values. The proposed analysis is based on a preliminary study of the Asymptotic Vanishing Damping system introduced by Su et al. in to modelNesterov's accelerated gradient method in a continuous setting. Novel improved convergence results are also shown for the solutions of this dynamical system, including the finite length of the trajectory under the aforementioned geometry conditions. |
| title | Strong Convergence of FISTA Iterates under H{ö}lderian and Quadratic Growth Conditions |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2407.17063 |