{"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/analyzing-the-speed-of-convergence-in","title":"Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods","arxiv_id":"2410.01382","date":"2024-10-02","proceeding":null,"authors":["Bennet Gebken"],"abstract":"The Goldstein $\\varepsilon$-subdifferential is a relaxed version of the Clarke subdifferential which has recently appeared in several algorithms for nonsmooth optimization. With it comes the notion of $(\\varepsilon,\\delta)$-critical points, which are points in which the element with the smallest norm in the $\\varepsilon$-subdifferential has norm at most $\\delta$. To obtain points that are critical in the classical sense, $\\varepsilon$ and $\\delta$ must vanish. In this article, we analyze at which speed the distance of $(\\varepsilon,\\delta)$-critical points to the minimum vanishes with respect to $\\varepsilon$ and $\\delta$. Afterwards, we apply our results to gradient sampling methods and perform numerical experiments. Throughout the article, we put a special emphasis on supporting the theoretical results with simple examples that visualize them.","url_abs":"https://arxiv.org/abs/2410.01382v2","url_pdf":"https://arxiv.org/pdf/2410.01382v2.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":"analyzing-the-speed-of-convergence-in","repo_url":"https://github.com/b-gebken/DGS","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"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}