{"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-heavy-ball-method-for-nonconvex","title":"Universal heavy-ball method for nonconvex optimization under Hölder continuous Hessians","arxiv_id":"2303.01073","date":"2023-03-02","proceeding":null,"authors":["Naoki Marumo","Akiko Takeda"],"abstract":"We propose a new first-order method for minimizing nonconvex functions with Lipschitz continuous gradients and H\\\"older continuous Hessians. The proposed algorithm is a heavy-ball method equipped with two particular restart mechanisms. It finds a solution where the gradient norm is less than $\\epsilon$ in $O(H_{\\nu}^{\\frac{1}{2 + 2 \\nu}} \\epsilon^{- \\frac{4 + 3 \\nu}{2 + 2 \\nu}})$ function and gradient evaluations, where $\\nu \\in [0, 1]$ and $H_{\\nu}$ are the H\\\"older exponent and constant, respectively. Our algorithm is $\\nu$-independent and thus universal; it automatically achieves the above complexity bound with the optimal $\\nu \\in [0, 1]$ without knowledge of $H_{\\nu}$. In addition, the algorithm does not require other problem-dependent parameters as input, including the gradient's Lipschitz constant or the target accuracy $\\epsilon$. Numerical results illustrate that the proposed method is promising.","url_abs":"https://arxiv.org/abs/2303.01073v2","url_pdf":"https://arxiv.org/pdf/2303.01073v2.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-heavy-ball-method-for-nonconvex","repo_url":"https://github.com/n-marumo/restarted-hb","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","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}