{"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/the-first-optimal-acceleration-of-high-order","title":"The First Optimal Acceleration of High-Order Methods in Smooth Convex Optimization","arxiv_id":"2205.09647","date":"2022-05-19","proceeding":null,"authors":["Dmitry Kovalev","Alexander Gasnikov"],"abstract":"In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\\Omega\\left(\\epsilon^{-2/(3p+1)}\\right)$ on the number of the $p$-th order oracle calls required by an algorithm to find an $\\epsilon$-accurate solution to the problem, where the $p$-th order oracle stands for the computation of the objective function value and the derivatives up to the order $p$. However, the existing state-of-the-art high-order methods of Gasnikov et al. (2019b); Bubeck et al. (2019); Jiang et al. (2019) achieve the oracle complexity $\\mathcal{O}\\left(\\epsilon^{-2/(3p+1)} \\log (1/\\epsilon)\\right)$, which does not match the lower bound. The reason for this is that these algorithms require performing a complex binary search procedure, which makes them neither optimal nor practical. We fix this fundamental issue by providing the first algorithm with $\\mathcal{O}\\left(\\epsilon^{-2/(3p+1)}\\right)$ $p$-th order oracle complexity.","url_abs":"https://arxiv.org/abs/2205.09647v1","url_pdf":"https://arxiv.org/pdf/2205.09647v1.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":"abstracts"},"code_links":[{"paper_slug":"the-first-optimal-acceleration-of-high-order","repo_url":"https://github.com/OPTAMI/OPTAMI","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[{"task_slug":"open-question","task_name":"Open-Ended Question Answering"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2205.09647","atlas_url":"https://app.syntology.ai/?focus=2205.09647","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}