  To jointly overcome the communication bottleneck and privacy leakage of
wireless federated learning (FL), this paper studies a differentially private
over-the-air federated averaging (DP-OTA-FedAvg) system with a limited sum
power budget. With DP-OTA-FedAvg, the gradients are aligned by an alignment
coefficient and aggregated over the air, and channel noise is employed to
protect privacy. We aim to improve the learning performance by jointly
designing the device scheduling, alignment coefficient, and the number of
aggregation rounds of federated averaging (FedAvg) subject to sum power and
privacy constraints. We first present the privacy analysis based on
differential privacy (DP) to quantify the impact of the alignment coefficient
on privacy preservation in each communication round. Furthermore, to study how
the device scheduling, alignment coefficient, and the number of the global
aggregation affect the learning process, we conduct the convergence analysis of
DP-OTA-FedAvg in the cases of convex and non-convex loss functions. Based on
these analytical results, we formulate an optimization problem to minimize the
optimality gap of the DP-OTA-FedAvg subject to limited sum power and privacy
budgets. The problem is solved by decoupling it into two sub-problems. Given
the number of communication rounds, we conclude the relationship between the
number of scheduled devices and the alignment coefficient, which offers a set
of potential optimal solution pairs of device scheduling and the alignment
coefficient. Thanks to the reduced search space, the optimal solution can be
efficiently obtained. The effectiveness of the proposed policy is validated
through simulations.
