{"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/convergence-of-shallow-relu-networks-on","title":"Convergence of Shallow ReLU Networks on Weakly Interacting Data","arxiv_id":"2502.16977","date":"2025-02-24","proceeding":null,"authors":["Léo Dana","Francis Bach","Loucas Pillaud-Vivien"],"abstract":"We analyse the convergence of one-hidden-layer ReLU networks trained by gradient flow on $n$ data points. Our main contribution leverages the high dimensionality of the ambient space, which implies low correlation of the input samples, to demonstrate that a network with width of order $\\log(n)$ neurons suffices for global convergence with high probability. Our analysis uses a Polyak-{\\L}ojasiewicz viewpoint along the gradient-flow trajectory, which provides an exponential rate of convergence of $\\frac{1}{n}$. When the data are exactly orthogonal, we give further refined characterizations of the convergence speed, proving its asymptotic behavior lies between the orders $\\frac{1}{n}$ and $\\frac{1}{\\sqrt{n}}$, and exhibiting a phase-transition phenomenon in the convergence rate, during which it evolves from the lower bound to the upper, and in a relative time of order $\\frac{1}{\\log(n)}$.","url_abs":"https://arxiv.org/abs/2502.16977v1","url_pdf":"https://arxiv.org/pdf/2502.16977v1.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":"convergence-of-shallow-relu-networks-on","repo_url":"https://github.com/leodana2000/Convergence-High-Dimension","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}