  This paper investigates energy-efficient transmission protocols in
relay-assisted federated learning (FL) setup within industrial subnetworks,
considering latency and power constraints. In the subnetworks, devices
collaborate to train a global model by transmitting their local models at the
edge-enabled primary access (pAP) directly or via secondary access points
(sAPs), which act as relays to optimize the training latency. We begin by
formulating the energy efficiency problem for our proposed transmission
protocol. Given its non-convex nature, we decompose it to minimize
computational and transmission energy separately. First, we introduce an
algorithm that categorizes devices into single-hop and two-hop groups to
decrease transmission energy and then selects associated sAPs. Subsequently, we
optimize the transmit power, aiming to maximize energy efficiency. To that end,
we propose a Sequential Parametric Convex Approximation (SPCA) method to
configure system parameters jointly. Simulation results show a 5% improvement
in convergence, significantly reduced outage, and at least a twofold savings in
total energy achieved by our proposed algorithm compared to single-hop
transmission.
