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Recent work of Erlingsson, Feldman, Mironov, Raghunathan, Talwar, andThakurta [EFMRTT19] demonstrates that random shuffling amplifies differentialprivacy guarantees of locally randomized data. Such amplification impliessubstantially stronger privacy guarantees for systems in which data iscontributed anonymously [BEMMRLRKTS17] and has lead to significant interest inthe shuffle model of privacy [CSUZZ19; EFMRTT19].  We show that random shuffling of $n$ data records that are input to$\varepsilon_0$-differentially private local randomizers results in an$(O((1-e^{-\varepsilon_0})\sqrt{\frac{e^{\varepsilon_0}\log(1/\delta)}{n}}),\delta)$-differentially private algorithm. This significantly improves overprevious work and achieves the asymptotically optimal dependence in$\varepsilon_0$. Our result is based on a new approach that is simpler thanprevious work and extends to approximate differential privacy with nearly thesame guarantees. Importantly, our work also yields an algorithm for derivingtighter bounds on the resulting $\varepsilon$ and $\delta$ as well as R\'enyidifferential privacy guarantees. We show numerically that our algorithm gets towithin a small constant factor of the optimal bound. As a direct corollary ofour analysis we derive a simple and nearly optimal algorithm for frequencyestimation in the shuffle model of privacy. We also observe that our resultimplies the first asymptotically optimal privacy analysis of noisy stochasticgradient descent that applies to sampling without replacement.

Hiding Among the Clones: A Simple and Nearly Optimal Analysis of Privacy Amplification by Shuffling