  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.
