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# Infinitely Divisible Noise in the Low Privacy Regime

*Proceedings of The 33rd International Conference on Algorithmic Learning Theory*, PMLR 167:881-909, 2022.

#### Abstract

Federated learning, in which training data is distributed among users and never shared, has emerged as a popular approach to privacy-preserving machine learning. Cryptographic techniques such as secure aggregation are used to aggregate contributions, like a model update, from all users. A robust technique for making such aggregates differentially private is to exploit \emph{infinite divisibility} of the Laplace distribution, namely, that a Laplace distribution can be expressed as a sum of i.i.d. noise shares from a Gamma distribution, one share added by each user. However, Laplace noise is known to have suboptimal error in the low privacy regime for $\varepsilon$-differential privacy, where $\varepsilon > 1$ is a large constant. In this paper we present the first infinitely divisible noise distribution for real-valued data that achieves $\varepsilon$-differential privacy and has expected error that decreases exponentially with $\varepsilon$.