  In this paper, we consider communication-efficient over-the-air federated
learning (FL), where multiple edge devices with non-independent and identically
distributed datasets perform multiple local iterations in each communication
round and then concurrently transmit their updated gradients to an edge server
over the same radio channel for global model aggregation using over-the-air
computation (AirComp). We derive the upper bound of the time-average norm of
the gradients to characterize the convergence of AirComp-assisted FL, which
reveals the impact of the model aggregation errors accumulated over all
communication rounds on convergence. Based on the convergence analysis, we
formulate an optimization problem to minimize the upper bound to enhance the
learning performance, followed by proposing an alternating optimization
algorithm to facilitate the optimal transceiver design for AirComp-assisted FL.
As the alternating optimization algorithm suffers from high computation
complexity, we further develop a knowledge-guided learning algorithm that
exploits the structure of the analytic expression of the optimal transmit power
to achieve computation-efficient transceiver design. Simulation results
demonstrate that the proposed knowledge-guided learning algorithm achieves a
comparable performance as the alternating optimization algorithm, but with a
much lower computation complexity. Moreover, both proposed algorithms
outperform the baseline methods in terms of convergence speed and test
accuracy.
