发表机构
Google; Google Research（谷歌; 谷歌研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了二叉树机制在差分隐私连续计数中的渐近最优性问题，证明了纯和近似差分隐私下的紧下界，并确立了该机制在标准参数范围内的最优性。
AI 中文摘要
二叉树机制是用于差分隐私连续计数的标准算法，但自其提出以来，其在纯差分隐私下的渐近最优性一直悬而未决。我们解决了这个问题。对于固定的 $0 < \varepsilon \le 1$，我们证明了最坏情况期望 $\ell_\infty$ 误差的渐近紧下界为 $\Omega(\log^2 n)$，以及平均和最大每坐标期望平方误差的渐近紧下界为 $\Omega(\log^3 n)$。这些下界适用于任意机制，即使整个数据流事先已知。在近似差分隐私下，当 $\delta\le n^{-c}$ 对于任意固定 $c>0$ 时，同样的下界也成立。我们的下界与使用拉普拉斯噪声的二叉树机制相匹配，确立了其在标准 $\delta \ll 1/n$ 范围内，在纯差分隐私和近似差分隐私下的渐近最优性。我们的证明使用了一个在树上具有有界指数分数的单一硬分布。对该分数进行简单修改，即可使同一框架为所有三种误差度量建立紧下界。
英文摘要
The Binary Tree Mechanism is a standard algorithm for differentially private continual counting, but its asymptotic optimality under pure differential privacy has remained unresolved since its introduction. We resolve this question. For fixed $0 < \varepsilon \le 1$, we prove asymptotically tight lower bounds of $Ω(\log^2 n)$ for worst-case expected $\ell_\infty$ error and $Ω(\log^3 n)$ for mean and maximum per-coordinate expected squared error. These bounds hold for arbitrary mechanisms, even when the entire stream is available in advance. The same lower bounds hold under approximate differential privacy whenever $δ\le n^{-c}$, for any fixed $c>0$. Our lower bounds match the Binary Tree Mechanism instantiated with Laplace noise, establishing its asymptotic optimality under both pure differential privacy and approximate differential privacy in the standard regime of $δ\ll1/n$. Our proof uses a single hard distribution with a bounded exponential score on a tree. A simple modification of the score allows the same framework to establish tight lower bounds for all three error measures.
