  Federated learning (FL) is an attractive paradigm for making use of rich
distributed data while protecting data privacy. Nonetheless, nonideal
communication links and limited transmission resources have become the
bottleneck of the implementation of fast and accurate FL. In this paper, we
study joint optimization of communications and FL based on analog aggregation
transmission in realistic wireless networks. We first derive a closed-form
expression for the expected convergence rate of FL over the air, which
theoretically quantifies the impact of analog aggregation on FL. Based on the
analytical result, we develop a joint optimization model for accurate FL
implementation, which allows a parameter server to select a subset of workers
and determine an appropriate power scaling factor. Since the practical setting
of FL over the air encounters unobservable parameters, we reformulate the joint
optimization of worker selection and power allocation using controlled
approximation. Finally, we efficiently solve the resulting mixed-integer
programming problem via a simple yet optimal finite-set search method by
reducing the search space. Simulation results show that the proposed solutions
developed for realistic wireless analog channels outperform a benchmark method,
and achieve comparable performance of the ideal case where FL is implemented
over noise-free wireless channels.
