Title: Shuffle Gaussian Mechanism for Differential Privacy. Abstract: We study Gaussian mechanism in the shuffle model of differential privacy (DP).
We present the \textit{first} non-trivial privacy guarantee of the mechanism by showing that its R{\'e}nyi differential privacy (RDP) is of the form:
$$
\epsilon(\lambda) =
% D_{\lambda}(\calM(D)||\calM(D')) =
\frac{1}{\lambda-1}\log\left(\frac{e^{-\lambda/2\sigma^2}}{n^\lambda}\sum_{\substack{k_1+\dotsc+k_n=\lambda;\\k_1,\dotsc,k_n\geq 0}}\binom{\lambda}{k_1,\dotsc,k_n}e^{\sum_{i=1}^nk_i^2/2\sigma^2}\right)
$$
We further prove that the RDP is strictly upper-bounded by the Gaussian RDP without shuffling.
The shuffle Gaussian RDP is advantageous in composing multiple DP mechanisms, where we demonstrate its improvement over the state-of-the-art approximate DP composition theorems in privacy guarantees of the shuffle model.
Our formalism also has immediate application in several problems studied in the literature, including learning with stochastic gradient descent and distributed/federated learning, of which an empirical study is presented to demonstrate the efficacy of learning privately while employing the shuffle Gaussian mechanism.