  Over-the-air federated edge learning (Air-FEEL) is a communication-efficient
solution for privacy-preserving distributed learning over wireless networks.
Air-FEEL allows "one-shot" over-the-air aggregation of gradient/model-updates
by exploiting the waveform superposition property of wireless channels, and
thus promises an extremely low aggregation latency that is independent of the
network size. However, such communication efficiency may come at a cost of
learning performance degradation due to the aggregation error caused by the
non-uniform channel fading over devices and noise perturbation. Prior work
adopted channel inversion power control (or its variants) to reduce the
aggregation error by aligning the channel gains, which, however, could be
highly suboptimal in deep fading scenarios due to the noise amplification. To
overcome this issue, we investigate the power control optimization for
enhancing the learning performance of Air-FEEL. Towards this end, we first
analyze the convergence behavior of the Air-FEEL by deriving the optimality gap
of the loss-function under any given power control policy. Then we optimize the
power control to minimize the optimality gap for accelerating convergence,
subject to a set of average and maximum power constraints at edge devices. The
problem is generally non-convex and challenging to solve due to the coupling of
power control variables over different devices and iterations. To tackle this
challenge, we develop an efficient algorithm by jointly exploiting the
successive convex approximation (SCA) and trust region methods. Numerical
results show that the optimized power control policy achieves significantly
faster convergence than the benchmark policies such as channel inversion and
uniform power transmission.
