  Federated Learning (FL) refers to distributed protocols that avoid direct raw
data exchange among the participating devices while training for a common
learning task. This way, FL can potentially reduce the information on the local
data sets that is leaked via communications. In order to provide formal privacy
guarantees, however, it is generally necessary to put in place additional
masking mechanisms. When FL is implemented in wireless systems via uncoded
transmission, the channel noise can directly act as a privacy-inducing
mechanism. This paper demonstrates that, as long as the privacy constraint
level, measured via differential privacy (DP), is below a threshold that
decreases with the signal-to-noise ratio (SNR), uncoded transmission achieves
privacy "for free", i.e., without affecting the learning performance. More
generally, this work studies adaptive power allocation (PA) for decentralized
gradient descent in wireless FL with the aim of minimizing the learning
optimality gap under privacy and power constraints. Both orthogonal multiple
access (OMA) and non-orthogonal multiple access (NOMA) transmission with
"over-the-air-computing" are studied, and solutions are obtained in closed form
for an offline optimization setting. Furthermore, heuristic online methods are
proposed that leverage iterative one-step-ahead optimization. The importance of
dynamic PA and the potential benefits of NOMA versus OMA are demonstrated
through extensive simulations.
