{"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/near-optimal-methods-for-minimizing-star","title":"Near-Optimal Methods for Minimizing Star-Convex Functions and Beyond","arxiv_id":"1906.11985","date":"2019-06-27","proceeding":null,"authors":["Oliver Hinder","Aaron Sidford","Nimit S. Sohoni"],"abstract":"In this paper, we provide near-optimal accelerated first-order methods for minimizing a broad class of smooth nonconvex functions that are strictly unimodal on all lines through a minimizer. This function class, which we call the class of smooth quasar-convex functions, is parameterized by a constant $\\gamma \\in (0,1]$, where $\\gamma = 1$ encompasses the classes of smooth convex and star-convex functions, and smaller values of $\\gamma$ indicate that the function can be \"more nonconvex.\" We develop a variant of accelerated gradient descent that computes an $\\epsilon$-approximate minimizer of a smooth $\\gamma$-quasar-convex function with at most $O(\\gamma^{-1} \\epsilon^{-1/2} \\log(\\gamma^{-1} \\epsilon^{-1}))$ total function and gradient evaluations. We also derive a lower bound of $\\Omega(\\gamma^{-1} \\epsilon^{-1/2})$ on the worst-case number of gradient evaluations required by any deterministic first-order method, showing that, up to a logarithmic factor, no deterministic first-order method can improve upon ours.","url_abs":"https://arxiv.org/abs/1906.11985v3","url_pdf":"https://arxiv.org/pdf/1906.11985v3.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":"near-optimal-methods-for-minimizing-star","repo_url":"https://github.com/nimz/quasar-convex-acceleration","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1906.11985","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1906.11985"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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