  In a federated learning scenario where multiple parties jointly learn a model
from their respective data, there exist two conflicting goals for the choice of
appropriate algorithms. On one hand, private and sensitive training data must
be kept secure as much as possible in the presence of \textit{semi-honest}
partners, while on the other hand, a certain amount of information has to be
exchanged among different parties for the sake of learning utility. Such a
challenge calls for the privacy-preserving federated learning solution, which
maximizes the utility of the learned model and maintains a provable privacy
guarantee of participating parties' private data.
  This article illustrates a general framework that a) formulates the trade-off
between privacy loss and utility loss from a unified information-theoretic
point of view, and b) delineates quantitative bounds of privacy-utility
trade-off when different protection mechanisms including Randomization,
Sparsity, and Homomorphic Encryption are used. It was shown that in general
\textit{there is no free lunch for the privacy-utility trade-off} and one has
to trade the preserving of privacy with a certain degree of degraded utility.
The quantitative analysis illustrated in this article may serve as the guidance
for the design of practical federated learning algorithms.
