Count on Your Elders: Laplace vs Gaussian Noise

TL;DR

It is argued 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.

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 $(\varepsilon,δ)$-differential privacy when $δ$ 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 $δ$ is sufficiently small it improves the mean squared error over the best possible $(\varepsilon,δ)$-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 $δ= O(T^{-0.92})$. Second, we show that the noise added by the Gaussian mechanism can always be replaced by Laplace noise of comparable variance for the same $(ε, δ)$-differential privacy guarantee, and in fact for sufficiently small $δ$ 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 $Ω(\log(T)/\varepsilon)$, matching the known lower bound for worst-case inputs.

Figures (2)

• Figure 1: Complete 5-ary tree with 25 leaves containing inputs $\mathbf{x}_1,\dots,\mathbf{x}_{25}$ and inner vertices containing subtree sums. To compute the sum of the first 24 inputs we can add 4 inner vertices and 4 leaves (shown in green) --- in general, the worst-case number of vertices for $k$-ary trees is $k-1$ per level. When subtree sums are made private by random (Laplace) noise the variance of a prefix sum estimator is proportional to the number of terms added. In \ref{['sec:kary_trees']} we analyze the privacy and utility of this natural generalization of the binary tree mechanism to $k$-ary trees, among other things showing that the mean number of terms is about half of the worst case.
• Figure 2: Complete ternary trees, each with 9 leaves containing inputs $x_1,\dots,x_{9}$ and inner vertices containing subtree sums. The trees illustrate two different possibilities for computing the sum of the first 8 inputs: (\ref{['fig: ter2']}) add two inner vertices and two leaf vertices, or (\ref{['fig: ter1']}) subtract one vertex (shown in red) from the sum stored in the root vertex (shown in green). The variance of using the latter method, with subtraction, is half of the former method, assuming the noise distribution for each vertex is the same. In \ref{['sec:ksub']} we analyze the privacy and utility of our new generalization of the binary tree mechanism to $k$-ary trees that makes use of subtraction. Among other things we show that the worst-case number of terms needed per level is $\lfloor k/2\rfloor$, and the mean number of terms is about half of the worst case.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Theorem 3: Approximate DP for the Laplace Mechanism
• Theorem 4
• Definition 5: $k$-ary trees
• Definition 6: $k$-ary representation of integer
• Lemma 7
• Lemma 8
• Theorem 9
• Definition 10: Offset $k$-ary representation of integers