Fast Optimal Locally Private Mean Estimation via Random Projections

TL;DR

This paper introduces ProjUnit, a simple and efficient framework for high-dimensional private mean estimation that achieves near-optimal error with low communication and computational costs, suitable for federated learning.

Contribution

The paper presents a novel framework, ProjUnit, that combines random projections with optimal algorithms to improve efficiency and accuracy in private mean estimation.

Findings

Achieves near-optimal error up to a 1+o(1) factor.

Reduces communication and computational costs significantly.

Demonstrates empirical utility comparable to optimal methods.

Abstract

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