发表机构
Aarhus University（奥胡斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在近似差分隐私下，连续计数问题的任何机制都必须有期望ℓ∞误差Ω(log^{3/2} n)，从而表明二叉树机制是渐近最优的。
AI 中文摘要
私有连续计数是差分隐私中的一个基本问题：给定一个长度为$n$的二进制流，其中每个$1$对应一个个体的贡献，目标是发布所有运行计数，同时保护每个个体的隐私。标准算法是二叉树机制，其高斯噪声变体在近似差分隐私下实现了与$\log^{3/2} n$成比例的期望$\ell_\infty$误差。这种对流长度的依赖是否必要一直是一个核心开放问题。在这项工作中，我们通过证明每个用于连续计数的差分隐私机制必须承受期望$\ell_\infty$误差$\Omega(\log^{3/2} n)$，解决了对$n$的依赖。这表明二叉树机制在近似差分隐私设置中是渐近最优的。作为推论，我们还获得了线性查询的遗传差异和私有$\ell_\infty$误差之间最大可能的分离，表明已知的关于遗传差异的一般上界具有对查询数量的最优依赖。
英文摘要
Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. For fixed privacy parameters, the standard binary tree mechanism achieves expected $\ell_\infty$ error $O(\log^{3/2} n)$ under approximate differential privacy and $O(\log^2 n)$ under pure differential privacy. Whether these dependences on the stream length are necessary has remained a central open problem. For fixed $\varepsilon\in(0,1)$, we prove a lower bound of $Ω(\log^{3/2} n)$ under approximate DP with sufficiently small fixed $δ>0$, and a lower bound of $Ω(\log^2 n)$ under pure DP. These bounds establish the optimality of the binary tree mechanism in both settings. The bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. Both proofs use the same decomposition and accumulation of residual noise along a tree. As a consequence of the approximate-DP bound, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.
