Count on Your Elders: Laplace vs Gaussian Noise

In recent years, Gaussian noise has become a popular tool in differentially private algorithms, often replacing Laplace noise which dominated the early literature on differential privacy. 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 when we seek to achieve $(\varepsilon,\delta)$-differential privacy for small values of $\delta$. 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 improves the mean squared error over all ``optimal'' $(\varepsilon,\delta)$-differentially private factorization mechanisms based on Gaussian noise whenever $\delta$ is sufficiently small. 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 $(\epsilon, \delta)$ 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.