{"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/propagation-of-chaos-in-one-hidden-layer","title":"Propagation of Chaos in One-hidden-layer Neural Networks beyond Logarithmic Time","arxiv_id":"2504.13110","date":"2025-04-17","proceeding":null,"authors":["Margalit Glasgow","Denny Wu","Joan Bruna"],"abstract":"We study the approximation gap between the dynamics of a polynomial-width neural network and its infinite-width counterpart, both trained using projected gradient descent in the mean-field scaling regime. We demonstrate how to tightly bound this approximation gap through a differential equation governed by the mean-field dynamics. A key factor influencing the growth of this ODE is the local Hessian of each particle, defined as the derivative of the particle's velocity in the mean-field dynamics with respect to its position. We apply our results to the canonical feature learning problem of estimating a well-specified single-index model; we permit the information exponent to be arbitrarily large, leading to convergence times that grow polynomially in the ambient dimension $d$. We show that, due to a certain ``self-concordance'' property in these problems -- where the local Hessian of a particle is bounded by a constant times the particle's velocity -- polynomially many neurons are sufficient to closely approximate the mean-field dynamics throughout training.","url_abs":"https://arxiv.org/abs/2504.13110v1","url_pdf":"https://arxiv.org/pdf/2504.13110v1.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":"propagation-of-chaos-in-one-hidden-layer","repo_url":"https://github.com/margalitglasgow/prop-chaos","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2504.13110","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}