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 (a.k.a. ROC curve) $f$ 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不等式。$f$-DP是一种以权衡函数（即ROC曲线）$f$表述的隐私概念，该函数限制了攻击者确定数据库中个体存在与否的能力。我们证明，Awan与Vadhan（2023）提出的规范噪声分布（CND）在半整数值处恰好达到反集中界，表明其尾部行为接近最优。同时，无论$f$-DP保证如何，所有CND均服从亚指数分布。对于对数凹CND，我们证明在具有相同隐私保证的任何噪声分布中，该类噪声在随机序意义下最小。针对整数值噪声，我们提出离散CND的新定义，证明离散CND总是存在且可通过对连续CND取整构造，且当为灵敏度为1的统计量设计时离散CND唯一。进一步证明，灵敏度为1的离散CND在随机序意义下小于其他整数值噪声。我们的理论结果揭示了$f$-DP框架下可能实现的多种隐私保证类型，并可整合到更复杂机制中以优化性能。