This paper studies federated learning for nonparametric regression in the context of distributed samples across different servers, each adhering to distinct differential privacy constraints. The setting we consider is heterogeneous, encompassing both varying sample sizes and differential privacy constraints across servers. Within this framework, both global and pointwise estimation are considered, and optimal rates of convergence over the Besov spaces are established. Distributed privacy-preserving estimators are proposed and their risk properties are investigated. Matching minimax lower bounds, up to a logarithmic factor, are established for both global and pointwise estimation. Together, these findings shed light on the tradeoff between statistical accuracy and privacy preservation. In particular, we characterize the compromise not only in terms of the privacy budget but also concerning the loss incurred by distributing data within the privacy framework as a whole. This insight captures the folklore wisdom that it is easier to retain privacy in larger samples, and explores the differences between pointwise and global estimation under distributed privacy constraints.

该论文研究了在分布式样本中进行非参数回归的联邦学习，其中每个样本服务器都遵守不同的差分隐私约束。在考虑到不同服务器的变化样本大小和差分隐私约束的异构设置中，该论文同时考虑了全局和逐点估计，并建立了在Besov空间上的收敛最优速率。通过提出分布式隐私保护估计器并研究其风险性质，该论文还建立了匹配的极小下界，对于全局和逐点估计，在对数因子上与之相符。这些发现揭示了统计准确性与隐私保护之间的权衡，特别是在整个隐私框架中分发数据所带来的损失，除了隐私预算之外。这种洞察力捕捉到了在更大的样本中更容易保持隐私的民间智慧，并探索了在分布式隐私约束下逐点估计和全局估计之间的差异。

具有异构分布式差分隐私约束的非参数回归的最优联邦学习