Joint Beamforming Design for Multifunctional RIS-Aided Over-the-Air Federated Learning

Over-the-air computation has emerged as a high-spectrum efficient and low-latency solution for model aggregation in federated learning (FL) by leveraging the superposition property of wireless channels. However, traditional over-the-air FL (AirFL) faces challenges such as signal misalignment, imperfect channel state information (CSI), and noisy fading channels. To enable more efficient and reliable AirFL in Internet of Things (IoT), we employ a multifunctional reconfigurable intelligent surface (MF-RIS) to alleviate mean square error (MSE) of AirFL model aggregation. By deriving the convergence analysis of AirFL in both convex and nonconvex settings, we unveil the impact of MSE on MF-RIS-aided AirFL under varying conditions. Based on the theoretical insights, we aim to minimize the MSE through a joint design of transceiver beamforming and MF-RIS coefficients, but it necessitates solving a mixed-integer nonlinear programming (MINLP) problem. To solve it efficiently, we propose an alternating optimization (AO) algorithm based on the semidefinite relaxation (SDR) approach and difference-of-convex (DC) programming. The efficacy of our approach is corroborated by numerical results, which underscore the performance gains achieved by the proposed algorithm. Additionally, the MF-RIS demonstrates remarkable proficiency in suppressing MSE and bolstering AirFL performance, even under conditions of imperfect CSI.