AI 中文总结
研究纯差分隐私连续计数误差界，提出通用矩阵分解机制改进最大平方误差和均方误差主导常数，给出算法实现，还证明特定分解类下界与上界渐近匹配，扩展下界至任意矩阵分解等问题待解。
AI 中文摘要
纯差分隐私下的连续计数是连续观察模型中最简单且研究最多的问题之一。然而，最大平方误差和均方误差的已知最佳上下界之间仍存在渐近差距。本文通过通用矩阵分解机制改进了最大平方误差和均方误差的主导常数，该机制从基于梯度优化获得的高质量低维分解出发，给出明确矩阵构造提升至任意大维度以改进误差保证，并提供了有效算法实现。同时证明了矩阵元素在{0,1}的分解类的下界为Ω(ε^(-2)log^3 n)，与该类上界渐近匹配。将此下界扩展到任意矩阵分解及其他机制仍是开放问题。
英文摘要
Continual counting under pure differential privacy is one of the simplest and most well-studied problems in the continual observation model. Nevertheless, an asymptotic gap remains between the best known upper and lower bounds for maximum squared error and mean squared error: the upper bound is $O(ε^{-2}\log^3 n)$, while the lower bound is $Ω(ε^{-2}\log^2 n)$, for both error metrics. The best known constant in the upper bound is achieved by the $k$-ary tree mechanism with the subtraction trick, due to Andersson, Pagh, Steiner, and Torkamani (FORC 2025). In this work, we improve the leading constant in the maximum squared error and the mean squared error. Our approach uses a general matrix factorization mechanism, yielding an improved bound for pure-DP continual counting that does not rely on a tree-based construction. The mechanism starts from a good-quality low-dimensional factorization, obtained via gradient-based optimization, and gives an explicit matrix construction that lifts this factorization to arbitrarily large dimensions, further improving its error guarantees. We offer an efficient algorithmic implementation of our mechanism. On the lower-bound side, we prove an $Ω(ε^{-2}\log^3 n)$ lower bound for the class of factorizations whose matrices have entries in $\{0,1\}$, matching the upper-bound asymptotics for this class. This class includes the binary tree mechanism and $k$-ary tree mechanisms without the subtraction trick. Extending this lower bound to arbitrary matrix factorizations, and beyond the matrix mechanism altogether, remains an open problem.
