  Federated learning has gained popularity as a means of training models
distributed across the wireless edge. The paper introduces delay-aware
hierarchical federated learning (DFL) to improve the efficiency of distributed
machine learning (ML) model training by accounting for communication delays
between edge and cloud. Different from traditional federated learning, DFL
leverages multiple stochastic gradient descent iterations on local datasets
within each global aggregation period and intermittently aggregates model
parameters through edge servers in local subnetworks. During global
synchronization, the cloud server consolidates local models with the outdated
global model using a local-global combiner, thus preserving crucial elements of
both, enhancing learning efficiency under the presence of delay. A set of
conditions is obtained to achieve the sub-linear convergence rate of O(1/k) for
strongly convex and smooth loss functions. Based on these findings, an adaptive
control algorithm is developed for DFL, implementing policies to mitigate
energy consumption and communication latency while aiming for sublinear
convergence. Numerical evaluations show DFL's superior performance in terms of
faster global model convergence, reduced resource consumption, and robustness
against communication delays compared to existing FL algorithms. In summary,
this proposed method offers improved efficiency and results when dealing with
both convex and non-convex loss functions.
