A common way to protect privacy of sensitive information is to introduce additional randomness, beyond sampling. Differential Privacy (DP) provides a rigorous framework for quantifying privacy risk of such procedures which allow for data summary releases, such as a statistic $T$. However in theory and practice, the structure of the statistic $T$ is often not carefully analyzed, resulting in inefficient implementation of DP mechanisms that introduce excessive randomness to minimize the risk, reducing the statistical utility of the result. We introduce the adjacent output space $S_T$ of $T$, and connect $S_T$ to the notion of sensitivity, which controls the amount of randomness required to protect privacy. Using $S_T$, we formalize the comparison of $K$-Norm Mechanisms and derive the optimal one as a function of the adjacent output space. We use these methods to extend the Objective Perturbation and the Functional mechanisms to arbitrary $K$-Mechanisms, and apply them to Logistic and Linear Regression, respectively, to allow for differentially private releases of statistical results. We compare the performance through simulations, and on a housing price data. Our results demonstrate that the choice of mechanism impacts the utility of the output, and our proposed methodology offers a significant improvement in utility for the same level of risk.

中文翻译：

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差分隐私中的结构和敏感性：比较 K 范数机制

保护敏感信息隐私的一种常见方法是引入额外的随机性，而不是抽样。差分隐私 (DP) 为量化此类程序的隐私风险提供了严格的框架，允许发布数据摘要，例如统计数据 $T$。然而，在理论和实践中，统计数据 $T$ 的结构往往没有被仔细分析，导致引入过多随机性以最小化风险的 DP 机制的低效实现，降低了结果的统计效用。我们引入了 $T$ 的相邻输出空间 $S_T$，并将 $S_T$ 连接到敏感度的概念，它控制保护隐私所需的随机性数量。使用$S_T$，我们将$K$-Norm Mechanisms 的比较形式化，并推导出作为相邻输出空间函数的最优值。我们使用这些方法将目标扰动和功能机制扩展到任意 $K$-机制，并将它们分别应用于逻辑回归和线性回归，以允许统计结果的差异化私有发布。我们通过模拟和房价数据来比较性能。我们的结果表明，机制的选择会影响输出的效用，并且我们提出的方法在相同风险水平的效用方面提供了显着改善。