Infinitely Divisible Noise for Differential Privacy: Nearly Optimal Error in the High 𝜀 Regime

Differential privacy can be achieved in a distributed manner, where multiple parties add independent noise such that their sum protects the overall dataset with differential privacy. A common technique here is for each party to sample their noise from the decomposition of an infinitely divisible distribution. We introduce two novel 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, which follows the sum of multiple i.i.d. discrete Laplace shares at different scales. The mechanisms can be parameterized to have 𝑂(Δ^3𝑒^{−𝜀}) and 𝑂 (min(Δ^3𝑒^{−𝜀}, Δ^2𝑒^{−2𝜀/3})) MSE, respectively, where the latter bound matches known optimality results. Furthermore, the MSDLap mechanism has the optimal MSE including constants as 𝜀 → ∞. 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 (Δ^2𝑒^{−2𝜀/3}) MSE. To our knowledge, these are the first infinitely divisible additive noise mechanisms that achieve order-optimal MSE under pure differential privacy for either the discrete or continuous setting, 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 Pagh and Stausholm’s Arete distribution which gives an MSE of 𝑂(Δ^2𝑒^{−𝜀/4}) [35]. We apply our results to improve a state of the art multi-message shuffle DP protocol from [3] in the high 𝜀 regime.