In this paper, we investigate federated contextual linear bandit learning within a wireless system that comprises a server and multiple devices. Each device interacts with the environment, selects an action based on the received reward, and sends model updates to the server. The primary objective is to minimize cumulative regret across all devices within a finite time horizon. To reduce the communication overhead, devices communicate with the server via over-the-air computation (AirComp) over noisy fading channels, where the channel noise may distort the signals. In this context, we propose a customized federated linear bandits scheme, where each device transmits an analog signal, and the server receives a superposition of these signals distorted by channel noise. A rigorous mathematical analysis is conducted to determine the regret bound of the proposed scheme. Both theoretical analysis and numerical experiments demonstrate the competitive performance of our proposed scheme in terms of regret bounds in various settings.
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