{"publication_identifier":{"isbn":["979-833131200-8"],"issn":["1071-9040"]},"month":"01","date_published":"2025-01-20T00:00:00Z","oa_version":"Preprint","project":[{"call_identifier":"H2020","grant_number":"101019564","_id":"bd9ca328-d553-11ed-ba76-dc4f890cfe62","name":"The design and evaluation of modern fully dynamic data structures"},{"_id":"34def286-11ca-11ed-8bc3-da5948e1613c","grant_number":"Z00422","name":"Efficient algorithms"},{"grant_number":"I05982","_id":"bda196b2-d553-11ed-ba76-8e8ee6c21103","name":"Static and Dynamic Hierarchical Graph Decompositions"},{"_id":"bd9e3a2e-d553-11ed-ba76-8aa684ce17fe","grant_number":"P33775","name":"Fast Algorithms for a Reactive Network Layer"}],"date_created":"2025-02-17T09:31:03Z","citation":{"ista":"Henzinger M, Upadhyay J. 2025. Improved differentially private continual observation using group algebra. Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 5, 2951–2970.","mla":"Henzinger, Monika, and Jalaj Upadhyay. “Improved Differentially Private Continual Observation Using Group Algebra.” <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol. 5, Association for Computing Machinery, 2025, pp. 2951–70, doi:<a href=\"https://doi.org/10.1137/1.9781611978322.95\">10.1137/1.9781611978322.95</a>.","apa":"Henzinger, M., &#38; Upadhyay, J. (2025). Improved differentially private continual observation using group algebra. In <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (Vol. 5, pp. 2951–2970). New Orleans, LA, United States: Association for Computing Machinery. <a href=\"https://doi.org/10.1137/1.9781611978322.95\">https://doi.org/10.1137/1.9781611978322.95</a>","short":"M. Henzinger, J. Upadhyay, in:, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, Association for Computing Machinery, 2025, pp. 2951–2970.","ama":"Henzinger M, Upadhyay J. Improved differentially private continual observation using group algebra. In: <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol 5. Association for Computing Machinery; 2025:2951-2970. doi:<a href=\"https://doi.org/10.1137/1.9781611978322.95\">10.1137/1.9781611978322.95</a>","chicago":"Henzinger, Monika, and Jalaj Upadhyay. “Improved Differentially Private Continual Observation Using Group Algebra.” In <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 5:2951–70. Association for Computing Machinery, 2025. <a href=\"https://doi.org/10.1137/1.9781611978322.95\">https://doi.org/10.1137/1.9781611978322.95</a>.","ieee":"M. Henzinger and J. Upadhyay, “Improved differentially private continual observation using group algebra,” in <i>Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, New Orleans, LA, United States, 2025, vol. 5, pp. 2951–2970."},"year":"2025","publication_status":"published","arxiv":1,"author":[{"full_name":"Henzinger, Monika H","last_name":"Henzinger","orcid":"0000-0002-5008-6530","id":"540c9bbd-f2de-11ec-812d-d04a5be85630","first_name":"Monika H"},{"first_name":"Jalaj","full_name":"Upadhyay, Jalaj","last_name":"Upadhyay"}],"_id":"19038","volume":5,"acknowledgement":"Monika Henzinger: This project has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (Grantagreement No. 101019564) and the Austrian Science Fund (FWF) grant DOI 10.55776/Z422,grant DOI 10.55776/I5982, and grant DOI 10.55776/P33775 with additional funding from the netidee SCIENCEStiftung, 2020–2024.Jalaj Upadhyay’s research was funded by the Rutgers Decanal Grant no. 302918 and an unrestricted giftfrom Google. This work was done in part while visiting the Institute of Science and Technology Austria (ISTA).The authors would like to thank Sarvagya Upadhyay for the initial discussion and feedback on the early draft of the paper. The authors would like to thank the anonymous reviewers, Brendan McMahan and Abhradeep Thakurta for the discussions that helped improve the presentation of the final version of the paper.","abstract":[{"lang":"eng","text":"Differentially private weighted prefix sum under continual observation is a crucial component in the production-level deployment of private next-word prediction for Gboard, which, according to Google, has over a billion users. More specifically, Google uses a differentially private mechanism to sum weighted gradients in its private follow-the-regularized leader algorithm. Apart from efficiency, the additive error of the private mechanism is crucial as multiplied with the square root of the model’s dimension d (with d ranging up to 10 trillion, for example, Switch Transformers or M6-10T), it determines the accuracy of the learning system. So, any improvement in leading constant matters significantly in practice. In this paper, we show a novel connection between mechanisms for continual weighted prefix sum and a concept in representation theory known as the group matrix introduced in correspondence between Dedekind and Frobenius (Sitzungsber. Preuss. Akad. Wiss. Berlin, 1897) and generalized by Schur (Journal für die reine und angewandte Mathematik, 1904). To the best of our knowledge, this is the first application of group algebra in the analysis of differentially private algorithms. Using this connection, we analyze a class of matrix norms known as factorization norms that give upper and lower bounds for the additive error under general ℓp-norms of the matrix mechanism. This allows us to give 1. the first efficient factorization that matches the best-known non-constructive upper bound on the factorization norm by Mathias (SIAM Journal of Matrix Analysis and Applications, 1993) for the matrix used in Google’s deployment, and also improves on the previous best-known constructive bound of Fichtenberger, Henzinger, and Upadhyay (ICML 2023) and Henzinger, Upadhyay, and Upadhyay (SODA 2023); thereby, partially resolving an open question in operator theory, 2. the first upper bound on the additive error for a large class of weight functions for weighted prefix sum problems, including the sliding window matrix (Bolot, Fawaz, Muthukrishnan, Nikolov, and Taft (ICDT 2013). We also improve the bound on factorizing the striped matrix used for outputting a synthetic graph that approximates all cuts (Fichtenberger, Henzinger, and Upadhyay (ICML 2023)); 3. a general improved upper bound on the factorization norms that depend on algebraic properties of the weighted sum matrices and that applies to a more general class of weighting functions than the ones considered in Henzinger, Upadhyay, and Upadhyay (SODA 2024). Using the known connection between these factorization norms and the ℓp-error of continual weighted sum, we give an upper bound on the ℓp-error for the continual weighted sum problem for p ≥ 2."}],"page":"2951 - 2970","department":[{"_id":"MoHe"}],"OA_place":"repository","date_updated":"2025-04-14T13:50:49Z","ec_funded":1,"scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2412.02840","open_access":"1"}],"publication":"Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms","title":"Improved differentially private continual observation using group algebra","intvolume":"         5","OA_type":"green","oa":1,"article_processing_charge":"No","quality_controlled":"1","language":[{"iso":"eng"}],"publisher":"Association for Computing Machinery","doi":"10.1137/1.9781611978322.95","day":"20","status":"public","type":"conference","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","conference":{"start_date":"2025-01-12","name":"SODA: Symposium on Discrete Algorithms","location":"New Orleans, LA, United States","end_date":"2025-01-15"},"external_id":{"arxiv":["2412.02840"]}}