{"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-scalable-estimate-of-the-extra-sample","title":"A scalable estimate of the extra-sample prediction error via approximate leave-one-out","arxiv_id":"1801.10243","date":"2018-01-30","proceeding":null,"authors":["Kamiar Rahnama Rad","Arian Maleki"],"abstract":"The paper considers the problem of out-of-sample risk estimation under the high dimensional settings where standard techniques such as $K$-fold cross validation suffer from large biases. Motivated by the low bias of the leave-one-out cross validation (LO) method, we propose a computationally efficient closed-form approximate leave-one-out formula (ALO) for a large class of regularized estimators. Given the regularized estimate, calculating ALO requires minor computational overhead. With minor assumptions about the data generating process, we obtain a finite-sample upper bound for $|\\text{LO} - \\text{ALO}|$. Our theoretical analysis illustrates that $|\\text{LO} - \\text{ALO}| \\rightarrow 0$ with overwhelming probability, when $n,p \\rightarrow \\infty$, where the dimension $p$ of the feature vectors may be comparable with or even greater than the number of observations, $n$. Despite the high-dimensionality of the problem, our theoretical results do not require any sparsity assumption on the vector of regression coefficients. Our extensive numerical experiments show that $|\\text{LO} - \\text{ALO}|$ decreases as $n,p$ increase, revealing the excellent finite sample performance of ALO. We further illustrate the usefulness of our proposed out-of-sample risk estimation method by an example of real recordings from spatially sensitive neurons (grid cells) in the medial entorhinal cortex of a rat.","url_abs":"https://arxiv.org/abs/1801.10243v4","url_pdf":"https://arxiv.org/pdf/1801.10243v4.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-scalable-estimate-of-the-extra-sample","repo_url":"https://github.com/Francis-Hsu/alocv","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}},{"paper_slug":"a-scalable-estimate-of-the-extra-sample","repo_url":"https://github.com/rnburn/peak-engines","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=1801.10243","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}