  Federated learning (FL) is a promising technique that enables many edge
devices to train a machine learning model collaboratively in wireless networks.
By exploiting the superposition nature of wireless waveforms, over-the-air
computation (AirComp) can accelerate model aggregation and hence facilitate
communication-efficient FL. Due to channel fading, power control is crucial in
AirComp. Prior works assume that the signals to be aggregated from each device,
i.e., local gradients have identical statistics. In FL, however, gradient
statistics vary over both training iterations and feature dimensions, and are
unknown in advance. This paper studies the power control problem for
over-the-air FL by taking gradient statistics into account. The goal is to
minimize the aggregation error by optimizing the transmit power at each device
subject to peak power constraints. We obtain the optimal policy in closed form
when gradient statistics are given. Notably, we show that the optimal transmit
power is continuous and monotonically decreases with the squared multivariate
coefficient of variation (SMCV) of gradient vectors. We then propose a method
to estimate gradient statistics with negligible communication cost.
Experimental results demonstrate that the proposed gradient-statistics-aware
power control achieves higher test accuracy than the existing schemes for a
wide range of scenarios.
