Count on Your Elders: Laplace vs Gaussian Noise

TL;DR

This paper compares Laplace and Gaussian noise in differential privacy, showing Laplace noise can outperform Gaussian in certain scenarios, especially for small delta, and introduces a new mechanism with improved accuracy.

Contribution

It introduces a novel $k$-ary binary tree mechanism using Laplace noise, improving privacy-accuracy trade-offs, and demonstrates Laplace noise's advantages over Gaussian noise for small delta in differential privacy.

Findings

Laplace noise can outperform Gaussian noise in $( extit{ε,δ})$-differential privacy for small δ.

A new $k$-ary binary tree mechanism with negative digits improves mean squared error.

Replacing Gaussian noise with Laplace noise of comparable variance is always possible for the same privacy guarantees.

Abstract

In recent years, Gaussian noise has become a popular tool in differentially private algorithms, often replacing Laplace noise which dominated the early literature. Gaussian noise is the standard approach to $\textit{approximate}$ differential privacy, often resulting in much higher utility than traditional (pure) differential privacy mechanisms. In this paper we argue that Laplace noise may in fact be preferable to Gaussian noise in many settings, in particular for $(\varepsilon,\delta)$-differential privacy when $\delta$ is small. We consider two scenarios: First, we consider the problem of counting under continual observation and present a new generalization of the binary tree mechanism that uses a $k$-ary number system with $\textit{negative digits}$ to improve the privacy-accuracy trade-off. Our mechanism uses Laplace noise and whenever $\delta$ is sufficiently small it improves the mean squared error over the best possible $(\varepsilon,\delta)$-differentially private factorization mechanisms based on Gaussian noise. Specifically, using $k=19$ we get an asymptotic improvement over the bound given in the work by Henzinger, Upadhyay and Upadhyay (SODA 2023) when $\delta = O(T^{-0.92})$. Second, we show that the noise added by the Gaussian mechanism can always be replaced by Laplace noise of comparable variance for the same $(\epsilon, \delta)$-differential privacy guarantee, and in fact for sufficiently small $\delta$ the variance of the Laplace noise becomes strictly better. This challenges the conventional wisdom that Gaussian noise should be used for high-dimensional noise. Finally, we study whether counting under continual observation may be easier in an average-case sense. We show that, under pure differential privacy, the expected worst-case error for a random input must be $\Omega(\log(T)/\varepsilon)$, matching the known lower bound for worst-case inputs.