Federated learning (FL) enables participating parties to collaboratively build a global model with boosted utility without disclosing private data information. Appropriate protection mechanisms have to be adopted to fulfill the opposing requirements in preserving \textit{privacy} and maintaining high model \textit{utility}. In addition, it is a mandate for a federated learning system to achieve high \textit{efficiency} in order to enable large-scale model training and deployment. We propose a unified federated learning framework that reconciles horizontal and vertical federated learning. Based on this framework, we formulate and quantify the trade-offs between privacy leakage, utility loss, and efficiency reduction, which leads us to the No-Free-Lunch (NFL) theorem for the federated learning system. NFL indicates that it is unrealistic to expect an FL algorithm to simultaneously provide excellent privacy, utility, and efficiency in certain scenarios. We then analyze the lower bounds for the privacy leakage, utility loss and efficiency reduction for several widely-adopted protection mechanisms including \textit{Randomization}, \textit{Homomorphic Encryption}, \textit{Secret Sharing} and \textit{Compression}. Our analysis could serve as a guide for selecting protection parameters to meet particular requirements.

翻译：联邦学习（FL）使参与方能够在无需泄露私有数据信息的情况下，协同构建具有更高效用的全局模型。为满足保护隐私与保持高模型效用这两类对立要求，必须采用适当的保护机制。此外，联邦学习系统还需实现高效率，以支持大规模模型的训练与部署。我们提出一个统一的联邦学习框架，该框架能够协调水平联邦学习与垂直联邦学习。基于此框架，我们形式化并量化了隐私泄露、效用损失与效率降低之间的权衡关系，进而推导出联邦学习系统的"无免费午餐"定理（No-Free-Lunch, NFL）。NFL指出，在某些场景下，期望联邦学习算法同时提供卓越的隐私、效用与效率是不切实际的。随后，我们分析了包括随机化（Randomization）、同态加密（Homomorphic Encryption）、秘密共享（Secret Sharing）和压缩（Compression）在内的几种广泛采用保护机制的隐私泄露、效用损失与效率降低的下界。我们的分析可作为选择保护参数以满足特定需求的指导。