Exact Worst-case Performance of First-order Methods for Composite Convex Optimization
| Autor: | Taylor, Adrien B., Hendrickx, Julien M., Glineur, François |
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| Rok vydání: | 2015 |
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| Zdroj: | SIAM Journal on Optimization, 27(3), 1283-1313 (2017) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1137/16M108104X |
| Popis: | We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this worst-case. We achieve this by reducing the computation of the worst-case to solving a convex semidefinite program, generalizing previous works on performance estimation by Drori and Teboulle [13] and the authors [43]. We use these developments to obtain a tighter analysis of the proximal point algorithm and of several variants of fast proximal gradient, conditional gradient, subgradient and alternating projection methods. In particular, we present a new analytical worst-case guarantee for the proximal point algorithm that is twice better than previously known, and improve the standard worst-case guarantee for the conditional gradient method by more than a factor of two. We also show how the optimized gradient method proposed by Kim and Fessler in [22] can be extended by incorporating a projection or a proximal operator, which leads to an algorithm that converges in the worst-case twice as fast as the standard accelerated proximal gradient method [2]. Comment: Published in SIOPT (updated version with corrected typo) Code available at https://github.com/AdrienTaylor/Performance-Estimation-Toolbox |
| Databáze: | arXiv |
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