Differentially private federated learning is crucial for maintaining privacy in distributed environments. This paper investigates the challenges of high-dimensional estimation and inference under the constraints of differential privacy. First, we study scenarios involving an untrusted central server, demonstrating the inherent difficulties of accurate estimation in high-dimensional problems. Our findings indicate that the tight minimax rates depends on the high-dimensionality of the data even with sparsity assumptions. Second, we consider a scenario with a trusted central server and introduce a novel federated estimation algorithm tailored for linear regression models. This algorithm effectively handles the slight variations among models distributed across different machines. We also propose methods for statistical inference, including coordinate-wise confidence intervals for individual parameters and strategies for simultaneous inference. Extensive simulation experiments support our theoretical advances, underscoring the efficacy and reliability of our approaches.

翻译：差分隐私联邦学习对于在分布式环境中维护隐私至关重要。本文研究了在差分隐私约束下高维估计与统计推断的挑战。首先，我们探讨了不可信中央服务器场景，揭示了高维问题中准确估计的内在困难。研究结果表明，即使存在稀疏性假设，最紧的极小化极大速率仍取决于数据的高维性。其次，我们考虑了可信中央服务器场景，并针对线性回归模型提出了一种新颖的联邦估计算法。该算法有效处理了分布在不同机器上的模型之间的细微差异。我们还提出了统计推断方法，包括单个参数的坐标置信区间以及同时推断策略。大量仿真实验支持了我们的理论进展，突出了所提方法的有效性与可靠性。