{"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/universal-average-case-optimality-of-polyak","title":"Universal Average-Case Optimality of Polyak Momentum","arxiv_id":"2002.04664","date":"2020-02-11","proceeding":null,"authors":["Damien Scieur","Fabian Pedregosa"],"abstract":"Polyak momentum (PM), also known as the heavy-ball method, is a widely used optimization method that enjoys an asymptotic optimal worst-case complexity on quadratic objectives. However, its remarkable empirical success is not fully explained by this optimality, as the worst-case analysis -- contrary to the average-case -- is not representative of the expected complexity of an algorithm. In this work we establish a novel link between PM and the average-case analysis. Our main contribution is to prove that any optimal average-case method converges in the number of iterations to PM, under mild assumptions. This brings a new perspective on this classical method, showing that PM is asymptotically both worst-case and average-case optimal.","url_abs":"http://arxiv.org/abs/2002.04664v3","url_pdf":"http://arxiv.org/pdf/2002.04664v3.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":"universal-average-case-optimality-of-polyak","repo_url":"https://github.com/acca3003/d2l-es","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"universal-average-case-optimality-of-polyak","repo_url":"https://github.com/d2l-ai/d2l-en","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"universal-average-case-optimality-of-polyak","repo_url":"https://github.com/d2l-ai/d2l-tr","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"universal-average-case-optimality-of-polyak","repo_url":"https://github.com/stabuev/d2l-ru","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"NOASSERTION"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2002.04664","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}