Infinitely Divisible Noise for Differential Privacy: Nearly Optimal Error in the High $\varepsilon$ Regime

Abstract: Differential privacy (DP) can be achieved in a distributed manner, where multiple parties add independent noise such that their sum protects the overall dataset with DP. A common technique here is for each party to sample their noise from the decomposition of an infinitely divisible distribution. We analyze two mechanisms in this setting: 1) the generalized discrete Laplace (GDL) mechanism, whose distribution (which is closed under summation) follows from differences of i.i.d. negative binomial shares, and 2) the multi-scale discrete Laplace (MSDLap) mechanism, a novel mechanism following the sum of multiple i.i.d. discrete Laplace shares at different scales. For $\varepsilon \geq 1$, our mechanisms can be parameterized to have $O\left(\Delta^3 e^{-\varepsilon}\right)$ and $O\left(\min\left(\Delta^3 e^{-\varepsilon}, \Delta^2 e^{-2\varepsilon/3}\right)\right)$ MSE, respectively, where $\Delta$ denote the sensitivity; the latter bound matches known optimality results. We also show a transformation from the discrete setting to the continuous setting, which allows us to transform both mechanisms to the continuous setting and thereby achieve the optimal $O\left(\Delta^2 e^{-2\varepsilon / 3}\right)$ MSE. To our knowledge, these are the first infinitely divisible additive noise mechanisms that achieve order-optimal MSE under pure DP, so our work shows formally there is no separation in utility when query-independent noise adding mechanisms are restricted to infinitely divisible noise. For the continuous setting, our result improves upon the Arete mechanism from [Pagh and Stausholm, ALT 2022] which gives an MSE of $O\left(\Delta^2 e^{-\varepsilon/4}\right)$. Furthermore, we give an exact sampler tuned to efficiently implement the MSDLap mechanism, and we apply our results to improve a state of the art multi-message shuffle DP protocol in the high $\varepsilon$ regime.