提出新方法在隐私保护下精准估算方差与协方差，尤其适合数据量也需保密的场景。
Optimal Variance and Covariance Estimation under Differential Privacy in the Add-Remove Model and Beyond
- 基于贝齐尔基函数设计新型隐私机制，兼顾数据量与统计量隐私
- 在高隐私强度下达到最优误差上限，理论证明其最坏情况表现最佳
- 不仅适用于方差协方差，还可推广到其他统计任务，实用性强
本文研究在添加-移除模型下的差分隐私方差与协方差估计问题。尽管交换模型已有广泛研究，添加-移除模型因需同时保护数据集规模而更具挑战性。为此，我们基于新颖的贝齐尔机制（Bézier mechanism）——一种利用伯恩斯坦基的矩发布框架——构建高效估计方法。我们证明所提机制在高隐私强度下为极小极大最优，通过建立新的极小极大下界实现。此外，超越最坏情形分析，我们研究实例级效用，发现贝齐尔基估计器始终优于其他机制。最后，展示贝齐尔机制在方差协方差之外的统计任务中亦具有效性。
原文摘要 · Abstract (English)
In this paper, we study the problem of estimating the variance and covariance of datasets under differential privacy in the add-remove model. While estimation in the swap model has been extensively studied in the literature, the add-remove model remains less explored and more challenging, as the dataset size must also be kept private. To address this issue, we develop efficient mechanisms for variance and covariance estimation based on the \emph{Bézier mechanism}, a novel moment-release framework that leverages Bernstein bases. We prove that our proposed mechanisms are minimax optimal in the high-privacy regime by establishing new minimax lower bounds. Moreover, beyond worst-case scenarios, we analyze instance-wise utility and show that the Bézier-based estimator consistently achieves better utility compared to alternative mechanisms. Finally, we demonstrate the effectiveness of the Bézier mechanism beyond variance and covariance estimation, showcasing its applicability to other statistical tasks.
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