  When implementing hierarchical federated learning over wireless networks,
scalability assurance and the ability to handle both interference and device
data heterogeneity are crucial. This work introduces a new two-level learning
method designed to address these challenges, along with a scalable over-the-air
aggregation scheme for the uplink and a bandwidth-limited broadcast scheme for
the downlink that efficiently use a single wireless resource. To provide
resistance against data heterogeneity, we employ gradient aggregations.
Meanwhile, the impact of uplink and downlink interference is minimized through
optimized receiver normalizing factors. We present a comprehensive mathematical
approach to derive the convergence bound for the proposed algorithm, applicable
to a multi-cluster wireless network encompassing any count of collaborating
clusters, and provide special cases and design remarks. As a key step to enable
a tractable analysis, we develop a spatial model for the setup by modeling
devices as a Poisson cluster process over the edge servers and rigorously
quantify uplink and downlink error terms due to the interference. Finally, we
show that despite the interference and data heterogeneity, the proposed
algorithm not only achieves high learning accuracy for a variety of parameters
but also significantly outperforms the conventional hierarchical learning
algorithm.
