差分隐私中的结构与敏感性：K-范数机制的比较
Structure and Sensitivity in Differential Privacy: Comparing K-Norm Mechanisms
AI总结：
该研究针对差分隐私下实值统计向量发布的噪声最小化问题，引入敏感性空间并提出含三种准则的K-范数机制比较框架，得出最优机制由敏感性空间凸包生成，扩展相关机制并应用于回归任务，实验验证其在同等风险下效用显著提升。
摘要：
差分隐私（DP）为抵御任意敌手的可证明隐私保护提供了框架，同时允许发布汇总统计量和合成数据。我们研究在差分隐私下，通过K-范数机制类发布含噪声的实值统计向量$T$（敏感数据的函数）的问题，目标是最小化为实现隐私所添加的噪声。首先，我们引入$T$的敏感性空间，将敏感性多胞体和敏感性凸包的概念扩展到任意统计量$T$的场景。随后，我们提出一个包含三种方法的框架来比较K-范数机制，以识别最优K-范数机制：1）随机占优的多元扩展，2）机制的熵，3）给定方向下的条件方差。在所有这些准则下，最优K-范数机制均由敏感性空间的凸包生成。利用我们的方法，我们扩展了目标扰动机制和函数机制，并将这些工具应用于逻辑回归和线性回归，实现统计结果的隐私发布。通过模拟实验以及在房价数据集上的应用，我们证明，在相同风险水平下，我们提出的方法在效用上有显著提升。
英文摘要：
Differential privacy (DP), provides a framework for provable privacy protection against arbitrary adversaries, while allowing the release of summary statistics and synthetic data. We address the problem of releasing a noisy real-valued statistic vector $T$, a function of sensitive data under DP, via the class of $K$-norm mechanisms with the goal of minimizing the noise added to achieve privacy. First, we introduce the sensitivity space of $T$, which extends the concepts of sensitivity polytope and sensitivity hull to the setting of arbitrary statistics $T$. We then propose a framework consisting of three methods for comparing the $K$-norm mechanisms: 1) a multivariate extension of stochastic dominance, 2) the entropy of the mechanism, and 3) the conditional variance given a direction, to identify the optimal $K$-norm mechanism. In all of these criteria, the optimal $K$-norm mechanism is generated by the convex hull of the sensitivity space. Using our methodology, we extend the objective perturbation and functional mechanisms and apply these tools to logistic and linear regression, allowing for private releases of statistical results. Via simulations and an application to a housing price dataset, we demonstrate that our proposed methodology offers a substantial improvement in utility for the same level of risk.
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