A Central Limit Theorem for Regularized M-Estimators

[2608.15034] A Central Limit Theorem for Regularized M-Estimators Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Statistics Theory arXiv:2608.15034 (math) [Submitted on 15 Aug 2026] Title: A Central Limit Theorem for Regularized M-Estimators Authors: Cosme Louart View a PDF of the paper titled A Central Limit Theorem for Regularized M-Estimators, by Cosme Louart View PDF HTML (experimental) Abstract: We prove a quantitative central limit theorem for linear functionals of regularized empirical-risk minimizers in the proportional-dimensional regime \(p=O(n)\). The data columns are independent, not necessarily identically distributed, and satisfy a uniform columnwise Poincaré inequality. Under uniform curvature and smoothness assumptions, and for a quadratic regularizer, we show that every nondegenerate statistic \(\sqrt n\,u^\top\hat\theta\), centered by its expectation and normalized by its standard deviation, converges to a standard normal random variable in Wasserstein distance, with rate \(O((\log n)^7n^{-1/4})\). The proof is based on moment and stability bounds for the minimizer, a second-order leave-one-out expansion, and a perturbative normal-approximation argument for functions of independent variables. We also prove the variance upper bound \(\Var(u^\top\hat\theta)\le C\norm{u}_2^2/n\), identifying the \(\sqrt n\) fluctuation scale. Comments: 62 pages Subjects: Statistics Theory (math.ST) ; Probability (math.PR) MSC classes: 62F12, 60F05, 62E20, 62J07 ACM classes: G.3 Cite as: arXiv:2608.15034 [math.ST] (or arXiv:2608.15034v1 [math.ST] for this version) https://doi.org/10.48550/arXiv.2608.15034 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cosme Louart [ view email ] [v1] Sat, 15 Aug 2026 04:38:32 UTC (51 KB) Full-text links: Access Paper: View a PDF of the paper titled A Central Limit Theorem for Regularized M-Estimators, by Cosme Louart View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2026-08 Change to browse by: math math.PR stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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