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We study the problem of locally private mean estimation of high-dimensionalvectors in the Euclidean ball. Existing algorithms for this problem eitherincur sub-optimal error or have high communication and/or run-time complexity.We propose a new algorithmic framework, ProjUnit, for private mean estimationthat yields algorithms that are computationally efficient, have lowcommunication complexity, and incur optimal error up to a $1+o(1)$-factor. Ourframework is deceptively simple: each randomizer projects its input to a randomlow-dimensional subspace, normalizes the result, and then runs an optimalalgorithm such as PrivUnitG in the lower-dimensional space. In addition, weshow that, by appropriately correlating the random projection matrices acrossdevices, we can achieve fast server run-time. We mathematically analyze theerror of the algorithm in terms of properties of the random projections, andstudy two instantiations. Lastly, our experiments for private mean estimationand private federated learning demonstrate that our algorithms empiricallyobtain nearly the same utility as optimal ones while having significantly lowercommunication and computational cost.

Fast Optimal Locally Private Mean Estimation via Random Projections