Imperfect CSI: A Key Factor of Uncertainty to Over-the-Air Federated Learning

Over-the-air computation (AirComp) has recently been identified as a prominent technique to enhance communication efficiency of wireless federated learning (FL). This letter investigates the impact of channel state information (CSI) uncertainty at the transmitter on an AirComp enabled FL (AirFL) system with the truncated channel inversion strategy. To characterize the performance of the AirFL system, the weight divergence with respect to the ideal aggregation is analytically derived to evaluate learning performance loss. We explicitly reveal that the weight divergence deteriorates as $\mathcal{O}(1/\rho^2)$ as the level of channel estimation accuracy $\rho$ vanishes, and also has a decay rate of $\mathcal{O}(1/K^2)$ with the increasing number of participating devices, $K$. Building upon our analytical results, we formulate the channel truncation threshold optimization problem to adapt to different $\rho$, which can be solved optimally. Numerical results verify the analytical results and show that a lower truncation threshold is preferred with more accurate CSI.