Optimized Power Control Design for Over-the-Air Federated Edge Learning

Over-the-air federated edge learning (Air-FEEL) has emerged as a communication-efficient solution to enable distributed machine learning over edge devices by using their data locally to preserve the privacy. By exploiting the waveform superposition property of wireless channels, Air-FEEL allows the “one-shot” over-the-air aggregation of gradient-updates to enhance the communication efficiency, but at the cost of a compromised learning performance due to the aggregation errors caused by channel fading and noise. This paper investigates the transmission power control to combat against such aggregation errors in Air-FEEL. Different from conventional power control designs (e.g., to minimize the individual mean squared error (MSE) of the over-the-air aggregation at each round), we consider a new power control design aiming at directly maximizing the convergence speed. Towards this end, we first analyze the convergence behavior of Air-FEEL (in terms of the optimality gap) subject to aggregation errors at different communication rounds. It is revealed that if the aggregation estimates are unbiased, then the training algorithm would converge exactly to the optimal point with mild conditions; while if they are biased, then the algorithm would converge with an error floor determined by the accumulated estimate bias over communication rounds. Next, building upon the convergence results, we optimize the power control to directly minimize the derived optimality gaps under the cases without and with unbiased aggregation constraints, subject to a set of average and maximum power constraints at individual edge devices. We transform both problems into convex forms, and obtain their structured optimal solutions, both appearing in a form of regularized channel inversion, by using the Lagrangian duality method. Finally, numerical results show that the proposed power control policies achieve significantly faster convergence for Air-FEEL, as compared with benchmark policies with fixed power transmission or conventional MSE minimization.