In this paper, we establish anti-concentration inequalities for additive noise mechanisms which achieve $f$-differential privacy ($f$-DP), a notion of privacy phrased in terms of a tradeoff function $f$ which limits the ability of an adversary to determine which individuals were in the database. We show that canonical noise distributions (CNDs), proposed by Awan and Vadhan (2023), match the anti-concentration bounds at half-integer values, indicating that their tail behavior is near-optimal. We also show that all CNDs are sub-exponential, regardless of the $f$-DP guarantee. In the case of log-concave CNDs, we show that they are the stochastically smallest noise compared to any other noise distributions with the same privacy guarantee. In terms of integer-valued noise, we propose a new notion of discrete CND and prove that a discrete CND always exists, can be constructed by rounding a continuous CND, and that the discrete CND is unique when designed for a statistic with sensitivity 1. We further show that the discrete CND at sensitivity 1 is stochastically smallest compared to other integer-valued noises. Our theoretical results shed light on the different types of privacy guarantees possible in the $f$-DP framework and can be incorporated in more complex mechanisms to optimize performance.

翻译：本文针对实现$f$-差分隐私（$f$-DP）的加性噪声机制建立了反集中性不等式。$f$-DP是一种基于权衡函数$f$的隐私定义，该函数限制了攻击者推断特定个体是否存在于数据库中的能力。我们证明，由Awan与Vadhan（2023）提出的典型噪声分布在半整数值处达到反集中性边界，表明其尾部行为接近最优。同时，我们证明所有典型噪声分布均服从亚指数分布，且该性质与$f$-DP保证无关。对于对数凹典型噪声分布，我们证明在具有相同隐私保证的所有噪声分布中，该类分布具有随机最小性。在整数值噪声方面，我们提出了离散典型噪声分布的新定义，并证明离散典型噪声分布始终存在，可通过连续典型噪声分布的取整构造获得，且在灵敏度为1的统计量设计中具有唯一性。进一步，我们证明灵敏度为1的离散典型噪声分布在所有整数值噪声中具有随机最小性。我们的理论成果揭示了$f$-DP框架中可能实现的不同类型隐私保证，并可被整合至更复杂的机制中以优化性能。