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.
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