  In this paper, we consider federated learning (FL) over a noisy fading
multiple access channel (MAC), where an edge server aggregates the local models
transmitted by multiple end devices through over-the-air computation (AirComp).
To realize efficient analog federated learning over wireless channels, we
propose an AirComp-based FedSplit algorithm, where a threshold-based device
selection scheme is adopted to achieve reliable local model uploading. In
particular, we analyze the performance of the proposed algorithm and prove that
the proposed algorithm linearly converges to the optimal solutions under the
assumption that the objective function is strongly convex and smooth. We also
characterize the robustness of proposed algorithm to the ill-conditioned
problems, thereby achieving fast convergence rates and reducing communication
rounds. A finite error bound is further provided to reveal the relationship
between the convergence behavior and the channel fading and noise. Our
algorithm is theoretically and experimentally verified to be much more robust
to the ill-conditioned problems with faster convergence compared with other
benchmark FL algorithms.
