  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 problem whether such discrete-valued
mechanisms provide any privacy protection. In this paper, we study the local
differential privacy guarantees of discrete-valued mechanisms with finite
output space through the lens of $f$-differential privacy (DP). More
specifically, we advance the existing literature by deriving tight $f$-DP
guarantees for a variety of discrete-valued mechanisms, including the binomial
noise and the binomial mechanisms that are proposed for privacy preservation,
and the sign-based methods that are proposed for data compression, in
closed-form expressions. We further investigate the amplification in privacy by
sparsification and propose a ternary stochastic compressor. By leveraging
compression for privacy amplification, we improve the existing methods by
removing the dependency of accuracy (in terms of mean square error) on
communication cost in the popular use case of distributed mean estimation,
therefore breaking the three-way tradeoff between privacy, communication, and
accuracy. Finally, we discuss the Byzantine resilience of the proposed
mechanism and its application in federated learning.
