conference paper published yes Monika H Henzinger author 540c9bbd-f2de-11ec-812d-d04a5be856300000-0002-5008-6530 Jalaj Upadhyay author Sarvagya Upadhyay author MoHe department SODA: Symposium on Discrete Algorithms The design and evaluation of modern fully dynamic data structures project Efficient algorithms project Static and Dynamic Hierarchical Graph Decompositions project Fast Algorithms for a Reactive Network Layer project We study the problem of maintaining a differentially private decaying sum under continual observation. We give a unifying framework and an efficient algorithm for this problem for any sufficiently smooth function. Our algorithm is the first differentially private algorithm that does not have a multiplicative error for polynomially decaying weights. Our algorithm improves on all prior works on differentially private decaying sums under continual observation and recovers exactly the additive error for the special case of continual counting from Henzinger et al. (SODA 2023) as a corollary. Our algorithm is a variant of the matrix mechanism whose error depends on the γ2 and γF norm of the underlying matrix. We give a constructive proof for an almost exact upper bound on the γ2 and γF norm and an almost tight lower bound on the γ2 norm for a large class of lower-triangular matrices. This is the first non-trivial lower bound for lower-triangular matrices whose non-zero entries are not all the same. It includes matrices for all continual decaying sums problems, resulting in an upper bound on the additive error of any differentially private decaying sums algorithm under continual observation. We also explore some implications of our result in discrepancy theory and operator algebra. Given the importance of the γ2 norm in computer science and the extensive work in mathematics, we believe our result will have further applications. Society for Industrial and Applied Mathematics2024Alexandria, VA, United States eng 2307.0897010.1137/1.9781611977912.38 2024995-1018 Henzinger M, Upadhyay J, Upadhyay S. A unifying framework for differentially private sums under continual observation. In: <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol 2024. Society for Industrial and Applied Mathematics; 2024:995-1018. doi:<a href="https://doi.org/10.1137/1.9781611977912.38">10.1137/1.9781611977912.38</a> Henzinger, M., Upadhyay, J., &#38; Upadhyay, S. (2024). A unifying framework for differentially private sums under continual observation. In <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (Vol. 2024, pp. 995–1018). Alexandria, VA, United States: Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/1.9781611977912.38">https://doi.org/10.1137/1.9781611977912.38</a> Henzinger, Monika, Jalaj Upadhyay, and Sarvagya Upadhyay. “A Unifying Framework for Differentially Private Sums under Continual Observation.” In <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 2024:995–1018. Society for Industrial and Applied Mathematics, 2024. <a href="https://doi.org/10.1137/1.9781611977912.38">https://doi.org/10.1137/1.9781611977912.38</a>. M. Henzinger, J. Upadhyay, and S. Upadhyay, “A unifying framework for differentially private sums under continual observation,” in <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Alexandria, VA, United States, 2024, vol. 2024, pp. 995–1018. M. Henzinger, J. Upadhyay, S. Upadhyay, in:, Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2024, pp. 995–1018. Henzinger, Monika, et al. “A Unifying Framework for Differentially Private Sums under Continual Observation.” <i>Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol. 2024, Society for Industrial and Applied Mathematics, 2024, pp. 995–1018, doi:<a href="https://doi.org/10.1137/1.9781611977912.38">10.1137/1.9781611977912.38</a>. Henzinger M, Upadhyay J, Upadhyay S. 2024. A unifying framework for differentially private sums under continual observation. Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 2024, 995–1018. 152532024-03-31T22:01:13Z2025-04-14T13:50:49Z