{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/accelerated-affine-invariant-convergence","title":"Accelerated Affine-Invariant Convergence Rates of the Frank-Wolfe Algorithm with Open-Loop Step-Sizes","arxiv_id":"2310.04096","date":"2023-10-06","proceeding":null,"authors":["Elias Wirth","Javier Pena","Sebastian Pokutta"],"abstract":"Recent papers have shown that the Frank-Wolfe algorithm (FW) with open-loop step-sizes exhibits rates of convergence faster than the iconic $\\mathcal{O}(t^{-1})$ rate. In particular, when the minimizer of a strongly convex function over a polytope lies in the relative interior of a feasible region face, the FW with open-loop step-sizes $\\eta_t = \\frac{\\ell}{t+\\ell}$ for $\\ell \\in \\mathbb{N}_{\\geq 2}$ has accelerated convergence $\\mathcal{O}(t^{-2})$ in contrast to the rate $\\Omega(t^{-1-\\epsilon})$ attainable with more complex line-search or short-step step-sizes. Given the relevance of this scenario in data science problems, research has grown to explore the settings enabling acceleration in open-loop FW. However, despite FW's well-known affine invariance, existing acceleration results for open-loop FW are affine-dependent. This paper remedies this gap in the literature by merging two recent research trajectories: affine invariance (Wirth et al., 2023b) and open-loop step-sizes (Pena, 2021). In particular, we extend all known non-affine-invariant convergence rates for FW with open-loop step-sizes to affine-invariant results.","url_abs":"https://arxiv.org/abs/2310.04096v3","url_pdf":"https://arxiv.org/pdf/2310.04096v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"accelerated-affine-invariant-convergence","repo_url":"https://github.com/zib-iol/affine_invariant_open_loop_fw","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}