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.

翻译：空中计算（AirComp）最近被认为是提升无线联邦学习（FL）通信效率的一项关键技术。本文研究了发射端信道状态信息（CSI）不确定性对采用截断信道反转策略的空中计算联邦学习（AirFL）系统的影响。为了描述AirFL系统的性能，我们解析推导了相对于理想聚合的权重偏差，以评估学习性能损失。我们明确揭示出，随着信道估计精度$\rho$的降低，权重偏差以$\mathcal{O}(1/\rho^2)$的阶数恶化；同时，随着参与设备数量$K$的增加，其衰减阶数为$\mathcal{O}(1/K^2)$。基于分析结果，我们构建了信道截断阈值优化问题以适应不同的$\rho$值，该问题可得到最优解。数值结果验证了理论分析，并表明在CSI更精确时，应优先选择较低的截断阈值。