{"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/fundamental-tradeoffs-between-memorization","title":"Fundamental tradeoffs between memorization and robustness in random features and neural tangent regimes","arxiv_id":"2106.02630","date":"2021-06-04","proceeding":null,"authors":["Elvis Dohmatob"],"abstract":"This work studies the (non)robustness of two-layer neural networks in various high-dimensional linearized regimes. We establish fundamental trade-offs between memorization and robustness, as measured by the Sobolev-seminorm of the model w.r.t the data distribution, i.e the square root of the average squared $L_2$-norm of the gradients of the model w.r.t the its input. More precisely, if $n$ is the number of training examples, $d$ is the input dimension, and $k$ is the number of hidden neurons in a two-layer neural network, we prove for a large class of activation functions that, if the model memorizes even a fraction of the training, then its Sobolev-seminorm is lower-bounded by (i) $\\sqrt{n}$ in case of infinite-width random features (RF) or neural tangent kernel (NTK) with $d \\gtrsim n$; (ii) $\\sqrt{n}$ in case of finite-width RF with proportionate scaling of $d$ and $k$; and (iii) $\\sqrt{n/k}$ in case of finite-width NTK with proportionate scaling of $d$ and $k$. Moreover, all of these lower-bounds are tight: they are attained by the min-norm / least-squares interpolator (when $n$, $d$, and $k$ are in the appropriate interpolating regime). All our results hold as soon as data is log-concave isotropic, and there is label-noise, i.e the target variable is not a deterministic function of the data / features. We empirically validate our theoretical results with experiments. Accidentally, these experiments also reveal for the first time, (iv) a multiple-descent phenomenon in the robustness of the min-norm interpolator.","url_abs":"https://arxiv.org/abs/2106.02630v1","url_pdf":"https://arxiv.org/pdf/2106.02630v1.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":"fundamental-tradeoffs-between-memorization","repo_url":"https://github.com/dohmatob/multiple-descent-robustness","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"memorization","task_name":"Memorization"}],"methods":[{"method_slug":"ntk","method_name":"NTK"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}