Smooth Sensitivity Revisited: Towards Optimality

TL;DR

The paper addresses how to optimally select noise for smooth sensitivity-based differential privacy. It introduces the PolyPlace distribution with parameters $s = \mathrm{SS}_q^\gamma(D)/\gamma$ and $\alpha = \varepsilon/\gamma$, proving that the mechanism $M(D) = q(D) + \operatorname{PolyPlace}(s, \alpha)$ achieves $\varepsilon$-DP and exhibits a variance that, as $\gamma/\varepsilon \to 0$, matches the Laplace mechanism scaled to the smooth sensitivity; furthermore, as $\gamma \to 0$ the distribution converges to Laplace, establishing a Laplace-limit optimality. It demonstrates substantial variance reductions relative to the prior Student's $T$-based approaches and extends the allowable range of the smoothing parameter, with broad applicability across smooth-sensitivity-enabled DP tasks. The work also situates PolyPlace within the broader landscape of DP methods, including inverse sensitivity mechanisms, highlighting its practical advantages in terms of ease of implementation and tighter utility bounds. Overall, PolyPlace provides a theoretically grounded, practically advantageous noise-adding mechanism for DP that leverages smooth sensitivity to achieve improved accuracy while preserving rigorous privacy guarantees.

Abstract

Smooth sensitivity is one of the most commonly used techniques for designing practical differentially private mechanisms. In this approach, one computes the smooth sensitivity of a given query $q$ on the given input $D$ and releases $q(D)$ with noise added proportional to this smooth sensitivity. One question remains: what distribution should we pick the noise from? In this paper, we give a new class of distributions suitable for the use with smooth sensitivity, which we name the PolyPlace distribution. This distribution improves upon the state-of-the-art Student's T distribution in terms of standard deviation by arbitrarily large factors, depending on a "smoothness parameter" $γ$, which one has to set in the smooth sensitivity framework. Moreover, our distribution is defined for a wider range of parameter $γ$, which can lead to significantly better performance. Moreover, we prove that the PolyPlace distribution converges for $γ\rightarrow 0$ to the Laplace distribution and so does its variance. This means that the Laplace mechanism is a limit special case of the PolyPlace mechanism. This implies that out mechanism is in a certain sense optimal for $γ\to 0$.

Figures (2)

• Figure 1: Comparison of the standard deviation of our and the other distributions that can be used with the smooth sensitivity framework. Laplace distribution when used with smooth sensitivity only provides approximate differential privacy; we assume here $\delta = 10^{-5}$. The shape parameters for the two other distributions are numerically selected so as to minimize their standard deviations. The standard deviations are calculated in \ref{['sec:stddev_of_others']}. The dashed line represents the standard deviation of the Laplace distribution which, however, needs to be used with the larger global sensitivity and not smooth sensitivity.
• Figure 2: A log-scale plot of selected members of the Laplace and $\operatorname{PolyPlace}$ families of distributions. The values of $\alpha$ and $s$ correspond (for $\mathop{\mathrm{SS}}\nolimits(D) = 1$ and $\varepsilon = 1$) to $\gamma \approx 0.91, \gamma = 2/3$, and $\gamma = 0.2$, respectively. Note that the distributions get closer to Laplace, which is explained by the convergence from \ref{['thm:main_theorem']}.

Theorems & Definitions (31)

• Theorem 1: Informal version of \ref{['thm:main_theorem']}
• Definition 2: Smooth sensitivity
• Definition 6
• Lemma 7
• Theorem 8
• proof
• Lemma 9
• proof
• Lemma 10
• proof