Fast convergence of Frank-Wolfe algorithms on polytopes
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913848285462528 |
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| author | Wirth, Elias Pena, Javier Pokutta, Sebastian |
| author_facet | Wirth, Elias Pena, Javier Pokutta, Sebastian |
| contents | We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_18789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fast convergence of Frank-Wolfe algorithms on polytopes Wirth, Elias Pena, Javier Pokutta, Sebastian Optimization and Control 90C25, 90C52 We provide a template to derive convergence rates for the following popular versions of the Frank-Wolfe algorithm on polytopes: vanilla Frank-Wolfe, Frank-Wolfe with away steps, Frank-Wolfe with blended pairwise steps, and Frank-Wolfe with in-face directions. Our template shows how the convergence rates follow from two affine-invariant properties of the problem, namely, error bound and extended curvature. These properties depend solely on the polytope and objective function but not on any affine-dependent object like norms. For each one of the above algorithms, we derive rates of convergence ranging from sublinear to linear depending on the degree of the error bound. |
| title | Fast convergence of Frank-Wolfe algorithms on polytopes |
| topic | Optimization and Control 90C25, 90C52 |
| url | https://arxiv.org/abs/2406.18789 |