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 \emph{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 factorization mechanism whose error depends on the $\gamma_2$ and $\gamma_F$ norm of the underlying matrix. We give a constructive proof for an almost exact upper bound on the $\gamma_2$ and $\gamma_F$ norm and an almost tight lower bound on the $\gamma_2$ 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 $\gamma_2$ norm in computer science and the extensive work in mathematics, we believe our result will have further applications.

翻译：我们研究了持续观察下维护差分隐私衰减求和的问题。我们针对任意充分光滑的函数，提出了一个统一框架及其高效算法。这是首个对多项式衰减权重不存在乘法误差的差分隐私算法。该算法改进了所有先前关于持续观察下差分隐私衰减求和的研究，并作为推论精确恢复了Henzinger等人（SODA 2023）中持续计数特例的加性误差。该算法是因子分解机制的一种变体，其误差取决于底层矩阵的γ₂范数与γ_F范数。针对一大类下三角矩阵，我们构造性地证明了γ₂范数与γ_F范数几乎精确的上界，并给出了γ₂范数的几乎紧下界。这是首个针对非零元素不完全相同的下三角矩阵的非平凡下界，涵盖所有持续衰减求和问题对应的矩阵，进而在持续观察下为任何差分隐私衰减求和算法的加性误差提供了上界。我们还探讨了该结果在差异理论和算子代数的若干应用。鉴于γ₂范数在计算机科学中的重要性以及数学领域的广泛研究，我们相信该结果将具有进一步的应用价值。