{"article_type":"original","publisher":"National Academy of Sciences","scopus_import":"1","publication":"Proceedings of the National Academy of Sciences","month":"04","acknowledgement":"This research was funded in whole, or in part, by the Austrian Science Fund (FWF) Grant number COE 12. For the purpose of open access, the author has applied a CC BY public copyright license to any Author Accepted Manuscript version arising from this submission. The authors were also supported by the 2019 Lopez-Loreta prize, and Simone Bombari was supported by a Google PhD fellowship. We thank Diyuan Wu, Edwige Cyffers, Francesco Pedrotti, Inbar Seroussi, Nikita P. Kalinin, Pietro Pelliconi, Roodabeh Safavi, Yizhe Zhu, and Zhichao Wang for helpful discussions.","abstract":[{"lang":"eng","text":"Differentially private gradient descent (DP-GD) is a popular algorithm to train deep learning models with provable guarantees on the privacy of the training data. In the last decade, the problem of understanding its performance cost with respect to standard GD has received remarkable attention from the research community, which formally derived upper bounds on the excess population risk  RP  in different learning settings. However, existing bounds typically degrade with over-parameterization, i.e., as the number of parameters  p  gets larger than the number of training samples  n  -- a regime which is ubiquitous in current deep-learning practice. As a result, the lack of theoretical insights leaves practitioners without clear guidance, leading some to reduce the effective number of trainable parameters to improve performance, while others use larger models to achieve better results through scale. In this work, we show that in the popular random features model with quadratic loss, for any sufficiently large  p , privacy can be obtained for free, i.e.,  |RP|=o(1) , not only when the privacy parameter  ε  has constant order, but also in the strongly private setting  ε=o(1) . This challenges the common wisdom that over-parameterization inherently hinders performance in private learning."}],"ddc":["000"],"article_number":"e2423072122","article_processing_charge":"Yes (in subscription journal)","has_accepted_license":"1","publication_status":"published","oa":1,"author":[{"last_name":"Bombari","first_name":"Simone","full_name":"Bombari, Simone","id":"ca726dda-de17-11ea-bc14-f9da834f63aa"},{"last_name":"Mondelli","orcid":"0000-0002-3242-7020","first_name":"Marco","id":"27EB676C-8706-11E9-9510-7717E6697425","full_name":"Mondelli, Marco"}],"status":"public","arxiv":1,"intvolume":"       122","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"19627","OA_place":"publisher","OA_type":"hybrid","language":[{"iso":"eng"}],"day":"15","pmid":1,"title":"Privacy for free in the overparameterized regime","quality_controlled":"1","file":[{"file_id":"19648","checksum":"1ac6f78e368d35a0cafb4d2d9bd63443","date_created":"2025-05-05T07:27:54Z","file_size":2328320,"success":1,"file_name":"2025_PNAS_Bombari.pdf","content_type":"application/pdf","date_updated":"2025-05-05T07:27:54Z","access_level":"open_access","relation":"main_file","creator":"dernst"}],"oa_version":"Published Version","doi":"10.1073/pnas.2423072122","date_published":"2025-04-15T00:00:00Z","publication_identifier":{"eissn":["1091-6490"],"issn":["0027-8424"]},"citation":{"chicago":"Bombari, Simone, and Marco Mondelli. “Privacy for Free in the Overparameterized Regime.” <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences, 2025. <a href=\"https://doi.org/10.1073/pnas.2423072122\">https://doi.org/10.1073/pnas.2423072122</a>.","short":"S. Bombari, M. Mondelli, Proceedings of the National Academy of Sciences 122 (2025).","apa":"Bombari, S., &#38; Mondelli, M. (2025). Privacy for free in the overparameterized regime. <i>Proceedings of the National Academy of Sciences</i>. National Academy of Sciences. <a href=\"https://doi.org/10.1073/pnas.2423072122\">https://doi.org/10.1073/pnas.2423072122</a>","ista":"Bombari S, Mondelli M. 2025. Privacy for free in the overparameterized regime. Proceedings of the National Academy of Sciences. 122(15), e2423072122.","ama":"Bombari S, Mondelli M. Privacy for free in the overparameterized regime. <i>Proceedings of the National Academy of Sciences</i>. 2025;122(15). doi:<a href=\"https://doi.org/10.1073/pnas.2423072122\">10.1073/pnas.2423072122</a>","ieee":"S. Bombari and M. Mondelli, “Privacy for free in the overparameterized regime,” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 15. National Academy of Sciences, 2025.","mla":"Bombari, Simone, and Marco Mondelli. “Privacy for Free in the Overparameterized Regime.” <i>Proceedings of the National Academy of Sciences</i>, vol. 122, no. 15, e2423072122, National Academy of Sciences, 2025, doi:<a href=\"https://doi.org/10.1073/pnas.2423072122\">10.1073/pnas.2423072122</a>."},"type":"journal_article","volume":122,"project":[{"name":"Prix Lopez-Loretta 2019 - Marco Mondelli","_id":"059876FA-7A3F-11EA-A408-12923DDC885E"},{"_id":"92099302-16d5-11f0-9cad-f9a785f54fbd","name":"Trustworthy Deep Learning Theory: Private Over-Parameterized Models and Robust LLMs"}],"file_date_updated":"2025-05-05T07:27:54Z","date_updated":"2025-06-11T13:18:31Z","external_id":{"arxiv":["2410.14787"],"pmid":["40215275"]},"corr_author":"1","date_created":"2025-04-27T22:02:13Z","year":"2025","issue":"15","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"department":[{"_id":"MaMo"}]}