Privacy and communication constraints are two major bottlenecks in federated learning (FL) and analytics (FA). We study the optimal accuracy of mean and frequency estimation (canonical models for FL and FA respectively) under joint communication and $(\varepsilon, \delta)$-differential privacy (DP) constraints. We show that in order to achieve the optimal error under $(\varepsilon, \delta)$-DP, it is sufficient for each client to send $\Theta\left( n \min\left(\varepsilon, \varepsilon^2\right)\right)$ bits for FL and $\Theta\left(\log\left( n\min\left(\varepsilon, \varepsilon^2\right) \right)\right)$ bits for FA to the server, where $n$ is the number of participating clients. Without compression, each client needs $O(d)$ bits and $\log d$ bits for the mean and frequency estimation problems respectively (where $d$ corresponds to the number of trainable parameters in FL or the domain size in FA), which means that we can get significant savings in the regime $ n \min\left(\varepsilon, \varepsilon^2\right) = o(d)$, which is often the relevant regime in practice. Our algorithms leverage compression for privacy amplification: when each client communicates only partial information about its sample, we show that privacy can be amplified by randomly selecting the part contributed by each client.

翻译：隐私和通信约束是联邦学习（FL）和分析（FA）中的两大瓶颈。我们研究了在联合通信和$(\varepsilon, \delta)$-差分隐私（DP）约束下，均值估计和频率估计（分别作为联邦学习和分析的标准模型）的最优精度。结果表明，为实现$(\varepsilon, \delta)$-DP下的最优误差，每个客户端向服务器发送$\Theta\left( n \min\left(\varepsilon, \varepsilon^2\right)\right)$比特（对于联邦学习）和$\Theta\left(\log\left( n\min\left(\varepsilon, \varepsilon^2\right) \right)\right)$比特（对于联邦分析）即可满足需求，其中$n$是参与客户端的数量。若无压缩，每个客户端在均值估计和频率估计问题中分别需要$O(d)$比特和$\log d$比特（其中$d$对应于联邦学习中的可训练参数数量或联邦分析中的域大小）。这意味着在$ n \min\left(\varepsilon, \varepsilon^2\right) = o(d)$的常见实际场景下，我们能够实现显著的通信节省。我们的算法利用压缩实现隐私放大：当每个客户端仅传输其样本的部分信息时，我们证明通过随机选择每个客户端贡献的部分，可以增强隐私保护效果。