{"paper":{"title":"Linear Convergence of the Frank-Wolfe Algorithm over Product Polytopes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"math.OC","authors_text":"David Mart\\'inez-Rubio, Elias Wirth, Francisco Criado, Gabriele Iommazzo, Sebastian Pokutta","submitted_at":"2025-05-16T13:50:55Z","abstract_excerpt":"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 $\\mu$-Polyak-{\\L}ojasiewicz, 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 effi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.11259","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.11259/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}