{"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/continuous-time-frank-wolfe-does-not-zig-zag","title":"Continuous Time Frank-Wolfe Does Not Zig-Zag, But Multistep Methods Do Not Accelerate","arxiv_id":"2106.05753","date":"2021-06-10","proceeding":null,"authors":["Zhaoyue Chen","Mokhwa Lee","Yifan Sun"],"abstract":"The Frank-Wolfe algorithm has regained much interest in its use in structurally constrained machine learning applications. However, one major limitation of the Frank-Wolfe algorithm is the slow local convergence property due to the zig-zagging behavior. We observe that this zig-zagging phenomenon can be viewed as an artifact of discretization, as when the method is viewed as an Euler discretization of a continuous time flow, that flow does not zig-zag. For this reason, we propose multistep Frank-Wolfe variants based on discretizations of the same flow whose truncation errors decay as $O(\\Delta^p)$, where $p$ is the method's order. This strategy \"stabilizes\" the method, and allows tools like line search and momentum to have more benefit. However, in terms of a convergence rate, our result is ultimately negative, suggesting that no Runge-Kutta-type discretization scheme can achieve a better convergence rate than the vanilla Frank-Wolfe method. We believe that this analysis adds to the growing knowledge of flow analysis for optimization methods, and is a cautionary tale on the ultimate usefulness of multistep methods.","url_abs":"https://arxiv.org/abs/2106.05753v2","url_pdf":"https://arxiv.org/pdf/2106.05753v2.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":"continuous-time-frank-wolfe-does-not-zig-zag","repo_url":"https://github.com/Mokhwalee/Continuous-Time-Frank-Wolfe-Does-not-Zig-Zag","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"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}