{"paper":{"title":"Curvature-Dependent Lower Bounds for Frank-Wolfe","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Frank-Wolfe cannot converge faster than order T to the minus p over p minus one on p-uniformly convex sets for p at least 3.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Christophe Roux, Jannis Halbey, Sebastian Pokutta","submitted_at":"2026-05-11T14:01:13Z","abstract_excerpt":"The Frank-Wolfe algorithm achieves a convergence rate of $\\mathcal{O}(1/T)$ for smooth convex optimization over compact convex domains, accelerating to $\\mathcal{O}(1/T^2)$ when both the objective and the feasible set are strongly convex. This acceleration extends beyond strong convexity: Kerdreux et al. (2021a) proved rates of $\\mathcal{O}(T^{-p/(p-1)})$ over $p$-uniformly convex feasible sets, a class that interpolates between strongly convex sets and more general curved domains such as $\\ell_p$ balls. In this work, we establish a matching $\\Omega(T^{-p/(p-1)})$ lower bound for every $p\\ge 3"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we establish a matching Ω(T^{-p/(p-1)}) lower bound for every p≥3 under exact line search or short steps, and extend the lower bound to objectives satisfying a Hölderian error bound.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis relies on the dynamics of Frank-Wolfe iterates on simple instances under the assumption that the feasible set is p-uniformly convex for p≥3 and the objective satisfies standard smoothness and convexity.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Matching Ω(T^{-p/(p-1)}) lower bounds are proven for Frank-Wolfe convergence on p-uniformly convex domains for p ≥ 3.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Frank-Wolfe cannot converge faster than order T to the minus p over p minus one on p-uniformly convex sets for p at least 3.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9ce6a113f47a50f0aa0c00983f4993ef359d6041a4e575a8778c674073831de6"},"source":{"id":"2605.10595","kind":"arxiv","version":2},"verdict":{"id":"a32f47b1-49db-4222-acd9-02972e1dea9c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T04:37:33.056171Z","strongest_claim":"we establish a matching Ω(T^{-p/(p-1)}) lower bound for every p≥3 under exact line search or short steps, and extend the lower bound to objectives satisfying a Hölderian error bound.","one_line_summary":"Matching Ω(T^{-p/(p-1)}) lower bounds are proven for Frank-Wolfe convergence on p-uniformly convex domains for p ≥ 3.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis relies on the dynamics of Frank-Wolfe iterates on simple instances under the assumption that the feasible set is p-uniformly convex for p≥3 and the objective satisfies standard smoothness and convexity.","pith_extraction_headline":"Frank-Wolfe cannot converge faster than order T to the minus p over p minus one on p-uniformly convex sets for p at least 3."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10595/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:40:53.433984Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:01:17.595274Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:09:07.804717Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"85b2bf7fb30e2818fc36c5cc45e9c025ef67b91bd60c6faf005d2e9f5a30ee55"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"353813df009ef0516a3bf47639049e72624edf4506bddd83e0cebd6a650d2a74"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}