Optimizing Noise for $f$-Differential Privacy via Anti-Concentration and Stochastic Dominance

Abstract

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.

Figures (6)

• Figure 1: Illustration of Lemma \ref{['lem:anti']}, and the connection to hypothesis testing. When testing $H_0: N$ versus $H_1: N+t$, and using the rejection region $(a+t/2,\infty)$, the type I error probability is illustrated by the vertically shaded red region and the type II error probability is given by the horizontally shaded blue region.
• Figure 2: Comparison of central probabilities for Tulap and Laplace distributions, as $\epsilon$ varies. Left: $P(|\cdot|\leq 1/2)$, Right: $P(|\cdot|\leq 1/4)$. See Example \ref{['ex:tulapLaplace']} for details.
• Figure 3: Left: Tradeoff function $C_1=T(\mathrm{Cauchy}(0,1),\mathrm{Cauchy}(1,1))$ as well as $f_{\epsilon_U,0}$ and $f_{\epsilon_L,0}$, from Example \ref{['ex:cauchy']}. Right: Comparison of $P(|N|\leq t)$ where $N\sim \mathrm{Cauchy}(0,1)$ or $N$ is a CND for $C_1$. Vertical line is at $t=1/2$ and horizontal line is at 1.
• Figure 4: Tradeoff function of $1$-GDP as well as the tradeoff function $T(N,N+1)$, where $N$ is the unique discrete CND for $1$-GDP at sensitivity 1.
• Figure 5: Discrete staircase distribution, which is a discrete CND for $(1,.05)$-DP at $\Delta=6$.

Theorems & Definitions (58)

• Definition 1: $f$-DP: dong2022gaussian
• Lemma 1
• Lemma 1
• Remark 2
• Theorem 3
• Remark 4
• Definition 5: Canonical Noise Distribution: awan2023canonical
• Lemma 6: awan2023canonical
• Proposition 7: CND construction: awan2023canonical
• Lemma 7