  We examine federated learning (FL) with over-the-air (OTA) aggregation, where
mobile users (MUs) aim to reach a consensus on a global model with the help of
a parameter server (PS) that aggregates the local gradients. In OTA FL, MUs
train their models using local data at every training round and transmit their
gradients simultaneously using the same frequency band in an uncoded fashion.
Based on the received signal of the superposed gradients, the PS performs a
global model update. While the OTA FL has a significantly decreased
communication cost, it is susceptible to adverse channel effects and noise.
Employing multiple antennas at the receiver side can reduce these effects, yet
the path-loss is still a limiting factor for users located far away from the
PS. To ameliorate this issue, in this paper, we propose a wireless-based
hierarchical FL scheme that uses intermediate servers (ISs) to form clusters at
the areas where the MUs are more densely located. Our scheme utilizes OTA
cluster aggregations for the communication of the MUs with their corresponding
IS, and OTA global aggregations from the ISs to the PS. We present a
convergence analysis for the proposed algorithm, and show through numerical
evaluations of the derived analytical expressions and experimental results that
utilizing ISs results in a faster convergence and a better performance than the
OTA FL alone while using less transmit power. We also validate the results on
the performance using different number of cluster iterations with different
datasets and data distributions. We conclude that the best choice of cluster
aggregations depends on the data distribution among the MUs and the clusters.
