Numerical vector aggregation plays a crucial role in privacy-sensitive applications, such as distributed gradient estimation in federated learning and statistical analysis of key-value data. In the context of local differential privacy, this study provides a tight minimax error bound of $O(\frac{ds}{n\epsilon^2})$, where $d$ represents the dimension of the numerical vector and $s$ denotes the number of non-zero entries. By converting the conditional/unconditional numerical mean estimation problem into a frequency estimation problem, we develop an optimal and efficient mechanism called Collision. In contrast, existing methods exhibit sub-optimal error rates of $O(\frac{d^2}{n\epsilon^2})$ or $O(\frac{ds^2}{n\epsilon^2})$. Specifically, for unconditional mean estimation, we leverage the negative correlation between two frequencies in each dimension and propose the CoCo mechanism, which further reduces estimation errors for mean values compared to Collision. Moreover, to surpass the error barrier in local privacy, we examine privacy amplification in the shuffle model for the proposed mechanisms and derive precisely tight amplification bounds. Our experiments validate and compare our mechanisms with existing approaches, demonstrating significant error reductions for frequency estimation and mean estimation on numerical vectors.

翻译：数值向量聚合在隐私敏感应用中扮演着关键角色，例如联邦学习中的分布式梯度估计以及键值数据的统计分析。在本地差分隐私的背景下，本研究提供了紧致的极小化误差界 $O(\frac{ds}{n\epsilon^2})$，其中 $d$ 表示数值向量的维度，$s$ 表示非零条目的数量。通过将有条件/无条件数值均值估计问题转化为频率估计问题，我们开发了一种最优且高效的机制，称为 Collision。相比之下，现有方法表现出次优的误差率 $O(\frac{d^2}{n\epsilon^2})$ 或 $O(\frac{ds^2}{n\epsilon^2})$。具体而言，对于无条件均值估计，我们利用每个维度中两个频率之间的负相关性，提出了 CoCo 机制，该机制相比 Collision 进一步降低了均值估计误差。此外，为突破本地隐私中的误差屏障，我们研究了混洗模型中对所提出机制的隐私放大效应，并推导出精确紧致的放大界。我们的实验验证了所提机制与现有方法的比较，结果表明在频率估计和数值向量均值估计中实现了显著的误差降低。