{"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/a-non-asymptotic-moreau-envelope-theory-for","title":"A Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models","arxiv_id":"2210.12082","date":"2022-10-21","proceeding":null,"authors":["Lijia Zhou","Frederic Koehler","Pragya Sur","Danica J. Sutherland","Nathan Srebro"],"abstract":"We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\\ell$ can control the test error under all Moreau envelopes of the loss $\\ell$. We use our finite-sample bound to directly recover the \"optimistic rate\" of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand's well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation.","url_abs":"https://arxiv.org/abs/2210.12082v1","url_pdf":"https://arxiv.org/pdf/2210.12082v1.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":"a-non-asymptotic-moreau-envelope-theory-for","repo_url":"https://github.com/zhoulijia/moreau-envelope","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"lemma","task_name":"LEMMA"}],"methods":[{"method_slug":"linear-regression","method_name":"Linear Regression"},{"method_slug":"test","method_name":"Test"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2210.12082","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}