  This paper presents a novel hierarchical federated learning algorithm within
multiple sets that incorporates quantization for communication-efficiency and
demonstrates resilience to statistical heterogeneity. Unlike conventional
hierarchical federated learning algorithms, our approach combines gradient
aggregation in intra-set iterations with model aggregation in inter-set
iterations. We offer a comprehensive analytical framework to evaluate its
optimality gap and convergence rate, comparing these aspects with those of
conventional algorithms. Additionally, we develop a problem formulation to
derive optimal system parameters in a closed-form solution. Our findings reveal
that our algorithm consistently achieves high learning accuracy over a range of
parameters and significantly outperforms other hierarchical algorithms,
particularly in scenarios with heterogeneous data distributions.
