Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912580167008256 |
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| author | Iommazzo, Gabriele Martínez-Rubio, David Criado, Francisco Wirth, Elias Pokutta, Sebastian |
| author_facet | Iommazzo, Gabriele Martínez-Rubio, David Criado, Francisco Wirth, Elias Pokutta, Sebastian |
| contents | We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $μ$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes Iommazzo, Gabriele Martínez-Rubio, David Criado, Francisco Wirth, Elias Pokutta, Sebastian Optimization and Control Machine Learning We study the linear convergence of Frank-Wolfe algorithms over product polytopes. We analyze two condition numbers for the product polytope, namely the \emph{pyramidal width} and the \emph{vertex-facet distance}, based on the condition numbers of individual polytope components. As a result, for convex objectives that are $μ$-Polyak-Łojasiewicz, we show linear convergence rates quantified in terms of the resulting condition numbers. We apply our results to the problem of approximately finding a feasible point in a polytope intersection in high-dimensions, and demonstrate the practical efficiency of our algorithms through empirical results. |
| title | Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2505.11259 |