{"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/escaping-saddle-points-in-zeroth-order","title":"Escaping saddle points in zeroth-order optimization: the power of two-point estimators","arxiv_id":"2209.13555","date":"2022-09-27","proceeding":null,"authors":["Zhaolin Ren","Yujie Tang","Na Li"],"abstract":"Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem may be high-dimensional and/or time-varying. Most problems in these applications are nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing $\\Omega(d)$ function valuations per iteration (with $d$ denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on $2m$ (for any $1 \\leq m \\leq d$) function evaluations per iteration can not only find $\\epsilon$-second order stationary points polynomially fast, but do so using only $\\tilde{O}\\left(\\frac{d}{m\\epsilon^{2}\\bar{\\psi}}\\right)$ function evaluations, where $\\bar{\\psi} \\geq \\tilde{\\Omega}\\left(\\sqrt{\\epsilon}\\right)$ is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.","url_abs":"https://arxiv.org/abs/2209.13555v3","url_pdf":"https://arxiv.org/pdf/2209.13555v3.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":"escaping-saddle-points-in-zeroth-order","repo_url":"https://github.com/rafflesintown/escape-saddle-points-2pt","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2209.13555","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}