发表机构
The Wharton School, University of Pennsylvania; School of Statistics, East China Normal University（宾夕法尼亚大学沃顿商学院; 华东师范大学统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对中心零集中差分隐私下的正则参数估计，提出精确渐近效率理论，通过直径约束信息区域和逆信息变分泛函刻画极小极大风险，并给出多种模型的最优程序。
AI 中文摘要
保护个人隐私已成为现代数据分析中一个核心且紧迫的问题，鉴于当前生成和处理的数据量巨大。在本文中，我们针对中心零集中差分隐私下的正则参数估计，发展了一套精确渐近效率的系统理论。在隐私机制中，主导对象是一个直径约束的信息区域：由直径至多为一的统计量生成的信息矩阵集合。对于加权二次损失，我们证明了精确的局部极小极大风险是该区域上的一个逆信息变分泛函。更一般地，一个混合信息区域在不同机制间产生统一的效率常数，涵盖隐私机制和经典Fisher效率。一种匹配估计器释放一个近乎最优有界统计量的经验均值，并加上高斯噪声，然后局部反转其总体矩映射。我们的理论与经典效率理论的不同之处在于其集值信息几何和依赖于损失的效率估计器。作为示例，我们为各种具体模型提供了闭式常数和最优程序，包括一维正则族、高斯均值、分类概率向量以及回归模型等。该理论也通过精确的参数重新缩放转移到高斯差分隐私。
英文摘要
Protecting individual privacy has become a central and urgent concern in modern data analysis, given the vast quantities of data now generated and processed. In this paper, we develop a systematic theory of exact asymptotic efficiency for regular parametric estimation under central zero-concentrated differential privacy. In the privacy regime, the governing object is a diameter-constrained information region: the set of information matrices generated by statistics with diameter at most one. For weighted quadratic loss, we show that the exact local minimax risk is an inverse information variational functional over this region. More generally, a mixed information region yields a unified efficiency constant across different regimes, covering the privacy regime and classical Fisher efficiency. A matching estimator releases the empirical mean of a nearly optimal bounded statistic with Gaussian noise and locally inverts its population moment map. Our theory differs from classical efficiency theory in its set-valued information geometry and loss-dependent efficient estimator. As examples, we provide closed-form constants and optimal procedures for various concrete models, including one-dimensional regular families, Gaussian means, categorical probability vectors, and regression models among others. The theory also transfers to Gaussian differential privacy through an exact parameter rescaling.
