We consider a federated data analytics problem in which a server coordinates the collaborative data analysis of multiple users with privacy concerns and limited communication capability. The commonly adopted compression schemes introduce information loss into local data while improving communication efficiency, and it remains an open question whether such discrete-valued mechanisms provide any privacy protection. Considering that differential privacy has become the gold standard for privacy measures due to its simple implementation and rigorous theoretical foundation, in this paper, we study the privacy guarantees of discrete-valued mechanisms with finite output space in the lens of $f$-differential privacy (DP). By interpreting the privacy leakage as a hypothesis testing problem, we derive the closed-form expression of the tradeoff between type I and type II error rates, based on which the $f$-DP guarantees of a variety of discrete-valued mechanisms, including binomial mechanisms, sign-based methods, and ternary-based compressors, are characterized. We further investigate the Byzantine resilience of binomial mechanisms and ternary compressors and characterize the tradeoff among differential privacy, Byzantine resilience, and communication efficiency. Finally, we discuss the application of the proposed method to differentially private stochastic gradient descent in federated learning.

翻译：考虑一个联邦数据分析问题，其中服务器协调多个具有隐私顾虑和有限通信能力的用户进行协作数据分析。广泛采用的压缩方案在提升通信效率的同时导致本地数据的信息损失，而此类离散值机制能否提供任何隐私保护仍是一个悬而未决的问题。考虑到差分隐私因实现简单且理论基础严谨已成为隐私度量标准的黄金准则，本文从$f$-差分隐私(DP)视角研究具有有限输出空间的离散值机制的隐私保证。通过将隐私泄露解释为假设检验问题，我们推导了第一类与第二类错误率权衡的闭式表达，并据此刻画了包括二项机制、基于符号的方法和三元压缩器在内的多种离散值机制的$f$-DP保证。我们进一步研究了二项机制和三元压缩器对拜占庭攻击的鲁棒性，揭示了差分隐私、拜占庭鲁棒性和通信效率之间的权衡关系。最后，讨论了所提方法在联邦学习差分隐私随机梯度下降中的应用。