{"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-interpolators-rapid-norm-growth-and-the","title":"Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization","arxiv_id":"2403.07264","date":"2024-03-12","proceeding":null,"authors":["Yutong Wang","Rishi Sonthalia","Wei Hu"],"abstract":"We study the generalization capability of nearly-interpolating linear regressors: $\\boldsymbol{\\beta}$'s whose training error $\\tau$ is positive but small, i.e., below the noise floor. Under a random matrix theoretic assumption on the data distribution and an eigendecay assumption on the data covariance matrix $\\boldsymbol{\\Sigma}$, we demonstrate that any near-interpolator exhibits rapid norm growth: for $\\tau$ fixed, $\\boldsymbol{\\beta}$ has squared $\\ell_2$-norm $\\mathbb{E}[\\|{\\boldsymbol{\\beta}}\\|_{2}^{2}] = \\Omega(n^{\\alpha})$ where $n$ is the number of samples and $\\alpha >1$ is the exponent of the eigendecay, i.e., $\\lambda_i(\\boldsymbol{\\Sigma}) \\sim i^{-\\alpha}$. This implies that existing data-independent norm-based bounds are necessarily loose. On the other hand, in the same regime we precisely characterize the asymptotic trade-off between interpolation and generalization. Our characterization reveals that larger norm scaling exponents $\\alpha$ correspond to worse trade-offs between interpolation and generalization. We verify empirically that a similar phenomenon holds for nearly-interpolating shallow neural networks.","url_abs":"https://arxiv.org/abs/2403.07264v1","url_pdf":"https://arxiv.org/pdf/2403.07264v1.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-interpolators-rapid-norm-growth-and-the","repo_url":"https://github.com/yutongwangumich/near-interpolators-figures","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"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}