  In federated learning (FL), reducing the communication overhead is one of the
most critical challenges since the parameter server and the mobile devices
share the training parameters over wireless links. With such consideration, we
adopt the idea of SignSGD in which only the signs of the gradients are
exchanged. Moreover, most of the existing works assume Channel State
Information (CSI) available at both the mobile devices and the parameter
server, and thus the mobile devices can adopt fixed transmission rates dictated
by the channel capacity. In this work, only the parameter server side CSI is
assumed, and channel capacity with outage is considered. In this case, an
essential problem for the mobile devices is to select appropriate local
processing and communication parameters (including the transmission rates) to
achieve a desired balance between the overall learning performance and their
energy consumption. Two optimization problems are formulated and solved, which
optimize the learning performance given the energy consumption requirement, and
vice versa. Furthermore, considering that the data may be distributed across
the mobile devices in a highly uneven fashion in FL, a stochastic sign-based
algorithm is proposed. Extensive simulations are performed to demonstrate the
effectiveness of the proposed methods.
