{"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/adaptive-open-loop-step-sizes-for-accelerated","title":"Adaptive Open-Loop Step-Sizes for Accelerated Convergence Rates of the Frank-Wolfe Algorithm","arxiv_id":"2505.09886","date":"2025-05-15","proceeding":null,"authors":["Elias Wirth","Javier Peña","Sebastian Pokutta"],"abstract":"Recent work has shown that in certain settings, the Frank-Wolfe algorithm (FW) with open-loop step-sizes $\\eta_t = \\frac{\\ell}{t+\\ell}$ for a fixed parameter $\\ell \\in \\mathbb{N},\\, \\ell \\geq 2$, attains a convergence rate faster than the traditional $O(t^{-1})$ rate. In particular, when a strong growth property holds, the convergence rate attainable with open-loop step-sizes $\\eta_t = \\frac{\\ell}{t+\\ell}$ is $O(t^{-\\ell})$. In this setting there is no single value of the parameter $\\ell$ that prevails as superior. This paper shows that FW with log-adaptive open-loop step-sizes $\\eta_t = \\frac{2+\\log(t+1)}{t+2+\\log(t+1)}$ attains a convergence rate that is at least as fast as that attainable with fixed-parameter open-loop step-sizes $\\eta_t = \\frac{\\ell}{t+\\ell}$ for any value of $\\ell \\in \\mathbb{N},\\,\\ell\\geq 2$. To establish our main convergence results, we extend our previous affine-invariant accelerated convergence results for FW to more general open-loop step-sizes of the form $\\eta_t = g(t)/(t+g(t))$, where $g:\\mathbb{N}\\to\\mathbb{R}_{\\geq 0}$ is any non-decreasing function such that the sequence of step-sizes $(\\eta_t)$ is non-increasing. This covers in particular the fixed-parameter case by choosing $g(t) = \\ell$ and the log-adaptive case by choosing $g(t) = 2+ \\log(t+1)$. To facilitate adoption of log-adaptive open-loop step-sizes, we have incorporated this rule into the {\\tt FrankWolfe.jl} software package.","url_abs":"https://arxiv.org/abs/2505.09886v1","url_pdf":"https://arxiv.org/pdf/2505.09886v1.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":"adaptive-open-loop-step-sizes-for-accelerated","repo_url":"https://github.com/ZIB-IOL/open_loop_fast","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}