  Federated learning has emerged as a popular technique for distributing
machine learning (ML) model training across the wireless edge. In this paper,
we propose two timescale hybrid federated learning (TT-HF), which is a hybrid
between the device-to-server communication paradigm in federated learning and
device-to-device (D2D) communications for model training. In TT-HF, during each
global aggregation interval, devices (i) perform multiple stochastic gradient
descent iterations on their individual datasets, and (ii) aperiodically engage
in consensus formation of their model parameters through cooperative,
distributed D2D communications within local clusters. With a new general
definition of gradient diversity, we formally study the convergence behavior of
TT-HF, resulting in new convergence bounds for distributed ML. We leverage our
convergence bounds to develop an adaptive control algorithm that tunes the step
size, D2D communication rounds, and global aggregation period of TT-HF over
time to target a sublinear convergence rate of O(1/t) while minimizing network
resource utilization. Our subsequent experiments demonstrate that TT-HF
significantly outperforms the current art in federated learning in terms of
model accuracy and/or network energy consumption in different scenarios where
local device datasets exhibit statistical heterogeneity.
