Convergence of Over-the-Air Federated Learning With Imperfect Channel Estimates: A Unified View

Abstract

Over-the-air computation-assisted federated learning (OTA-FL) exploits the superposition property of the wireless channel to markedly reduce the latency and bandwidth requirements of federated learning. Devices adapt their transmit power to enable over-the-air aggregation of the local models. However, imperfect channel state information (CSI) and power constraints distort the aggregated model update at the receiver and affect the learning algorithm. We provide a novel, comprehensive analysis of OTA-FL schemes, which encompasses scaled-down channel inversion (SCI), truncated channel inversion (TCI), and controlled descent algorithm (CDA). Unlike prior studies that assume bounded estimation errors, we study a more realistic model in which the estimation error has unbounded support. Our analysis addresses both fixed and adaptive learning rates. While prior works on imperfect CSI focus only on convergence in expectation for a specific scheme and assume fixed learning rates, we provide technically stronger almost sure convergence guarantees for multiple schemes when the number of devices is large. Extensive experiments on linear regression and CIFAR-10 classification validate our theory even for a small number of devices. Adapting the learning rate leads to convergence even with very noisy estimates.