{"paper":{"title":"Stochastic Compositional Optimization via Hybrid Momentum Frank--Wolfe","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A hybrid momentum Frank-Wolfe algorithm achieves the optimal O(K^{-1/4}) convergence rate for stochastic compositional problems with merely Lipschitz outer functions.","cross_cats":["cs.LG"],"primary_cat":"math.OC","authors_text":"El Mahdi Chayti","submitted_at":"2026-05-14T19:20:22Z","abstract_excerpt":"Stochastic compositional optimization minimizes objectives of the form $\\min_{\\bm{x} \\in \\mathcal{X}} F(\\bm{f}(\\bm{x}), \\bm{x})$, where $\\bm{f}$ is accessible only through noisy stochastic queries. Existing methods for this problem assume that the outer function $F$ is continuously differentiable, which excludes many practically important applications such as robust max-of-losses, Conditional Value-at-Risk, and norm regularizers. We propose the Hybrid Momentum Stochastic Frank--Wolfe algorithm, which drops the smoothness assumption on $F$. By combining a momentum-based Jacobian tracker with a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish an O(K^{-1/4}) convergence rate in the generalized Frank-Wolfe gap for non-convex objectives with L_F-Lipschitz outer functions, matching the optimal complexity for projection-free single-sample stochastic methods under expected smoothness.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The outer function F is L_F-Lipschitz (but not necessarily differentiable), and the stochastic oracles satisfy expected smoothness or bounded r-th moments for r in (1,2]. This premise enters in the convergence analysis that bounds the generalized Frank-Wolfe gap.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Hybrid Momentum Stochastic Frank-Wolfe algorithm achieves O(K^{-1/4}) convergence in the generalized Frank-Wolfe gap for non-convex stochastic compositional optimization with Lipschitz outer functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A hybrid momentum Frank-Wolfe algorithm achieves the optimal O(K^{-1/4}) convergence rate for stochastic compositional problems with merely Lipschitz outer functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8ce409d1b17599da335ddb602a40b9f7d93d3a933105c4f71ea32188ab37594d"},"source":{"id":"2605.15350","kind":"arxiv","version":1},"verdict":{"id":"4f28231b-3d25-42ca-88ee-687410a79638","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T15:13:15.353263Z","strongest_claim":"We establish an O(K^{-1/4}) convergence rate in the generalized Frank-Wolfe gap for non-convex objectives with L_F-Lipschitz outer functions, matching the optimal complexity for projection-free single-sample stochastic methods under expected smoothness.","one_line_summary":"The Hybrid Momentum Stochastic Frank-Wolfe algorithm achieves O(K^{-1/4}) convergence in the generalized Frank-Wolfe gap for non-convex stochastic compositional optimization with Lipschitz outer functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The outer function F is L_F-Lipschitz (but not necessarily differentiable), and the stochastic oracles satisfy expected smoothness or bounded r-th moments for r in (1,2]. This premise enters in the convergence analysis that bounds the generalized Frank-Wolfe gap.","pith_extraction_headline":"A hybrid momentum Frank-Wolfe algorithm achieves the optimal O(K^{-1/4}) convergence rate for stochastic compositional problems with merely Lipschitz outer functions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15350/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T15:31:17.897075Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:23:17.789332Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.203456Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.749798Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"235384f2c8d7af225b33db293f624ed2eb7929872640ee8cbf2c49a7958816fd"},"references":{"count":32,"sample":[{"doi":"","year":2023,"title":"Conference on Learning Theory , pages=","work_id":"24de1194-bb5e-47eb-8b1b-a040d5e2566c","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"2024 , url =","work_id":"c28d2d3e-2633-40c3-a165-53d276975c5f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"IEEE Transactions on Signal Processing , volume=","work_id":"a95358f8-77ef-4467-aada-4aeb43748f8b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Mathematical Programming , volume=","work_id":"644b16c4-500b-46ab-a84e-70145964882b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Journal of Machine Learning Research , volume=","work_id":"aa7974ef-7ece-4a5e-af49-5643c7012368","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":32,"snapshot_sha256":"ea5f7c2b2495d6d74541803e5891a171dff3dd6dc8d55753e60a39f6bb868550","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6bd20c45fbe31e473231adc5f5fda970335c5cbf588ba2132bdbe6ba8dd41b8f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}