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

TL;DR

This paper analyzes how imperfect channel state information (CSI) affects over-the-air federated learning, deriving the impact on learning performance and optimizing channel truncation strategies.

Contribution

It provides an analytical characterization of weight divergence under CSI uncertainty and proposes an optimal truncation threshold adaptation method.

Findings

Weight divergence worsens as CSI accuracy decreases, following an O(1/ρ^2) rate.

Increasing the number of devices reduces divergence at an O(1/K^2) rate.

Optimal truncation thresholds depend on CSI accuracy, with lower thresholds for more accurate CSI.

Abstract

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.