  In this paper, we consider the problem of wireless federated learning based
on sign stochastic gradient descent (signSGD) algorithm via a multiple access
channel. When sending locally computed gradient's sign information, each mobile
device requires to apply precoding to circumvent wireless fading effects. In
practice, however, acquiring perfect knowledge of channel state information
(CSI) at all mobile devices is infeasible. In this paper, we present a simple
yet effective precoding method with limited channel knowledge, called
sign-alignment precoding. The idea of sign-alignment precoding is to protect
sign-flipping errors from wireless fadings. Under the Gaussian prior assumption
on the local gradients, we also derive the mean squared error (MSE)-optimal
aggregation function called Bayesian over-the-air computation (BayAirComp). Our
key finding is that one-bit precoding with BayAirComp aggregation can provide a
better learning performance than the existing precoding method even using
perfect CSI with AirComp aggregation.
