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.

翻译：我们研究了在欧几里得球中高维向量的局部差分隐私均值估计问题。现有算法在处理该问题时，或产生次优误差，或具有较高的通信和/或运行时复杂度。我们提出了一种新的算法框架ProjUnit，用于隐私均值估计，该框架所生成的算法计算效率高、通信复杂度低，且误差可优化至$1+o(1)$因子范围内。我们的框架看似简单：每个随机化器将其输入投影至随机低维子空间，对结果进行归一化，然后在低维空间中运行PrivUnitG等最优算法。此外，我们证明通过适当关联各设备间的随机投影矩阵，可以实现快速的服务器端运行时。我们基于随机投影的性质对算法误差进行了数学分析，并研究了两种具体实例。最后，针对隐私均值估计和隐私联邦学习的实验表明，我们的算法在获得与最优算法几乎相同效用的同时，显著降低了通信和计算成本。