Edge federated learning (FL) is an emerging paradigm that trains a global parametric model from distributed datasets based on wireless communications. This paper proposes a unit-modulus over-the-air computation (UMAirComp) framework to facilitate efficient edge federated learning, which simultaneously uploads local model parameters and updates global model parameters via analog beamforming. The proposed framework avoids sophisticated baseband signal processing, leading to low communication delays and implementation costs. Training loss bounds of UMAirComp FL systems are derived and two low-complexity large-scale optimization algorithms, termed penalty alternating minimization (PAM) and accelerated gradient projection (AGP), are proposed to minimize the nonconvex nonsmooth loss bound. Simulation results show that the proposed UMAirComp framework with PAM algorithm achieves a smaller mean square error of model parameters' estimation, training loss, and test error compared with other benchmark schemes. Moreover, the proposed UMAirComp framework with AGP algorithm achieves satisfactory performance while reduces the computational complexity by orders of magnitude compared with existing optimization algorithms. Finally, we demonstrate the implementation of UMAirComp in a vehicle-to-everything autonomous driving simulation platform. It is found that autonomous driving tasks are more sensitive to model parameter errors than other tasks since the neural networks for autonomous driving contain sparser model parameters.
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