This paper investigates the problem of model aggregation in federated learning systems aided by multiple reconfigurable intelligent surfaces (RISs). The effective integration of computation and communication is achieved by over-the-air computation (AirComp). Since all local parameters are transmitted over shared wireless channels, the undesirable propagation error inevitably deteriorates the performance of global aggregation. The objective of this work is to 1) reduce the signal distortion of AirComp; 2) enhance the convergence rate of federated learning. Thus, the mean-square-error and the device set are optimized by designing the transmit power, controlling the receive scalar, tuning the phase shifts, and selecting participants in the model uploading process. The formulated mixed-integer non-linear problem (P0) is decomposed into a non-convex problem (P1) with continuous variables and a combinatorial problem (P2) with integer variables. To solve subproblem (P1), the closed-form expressions for transceivers are first derived, then the multi-antenna cases are addressed by the semidefinite relaxation. Next, the problem of phase shifts design is tackled by invoking the penalty-based successive convex approximation method. In terms of subproblem (P2), the difference-of-convex programming is adopted to optimize the device set for convergence acceleration, while satisfying the aggregation error demand. After that, an alternating optimization algorithm is proposed to find a suboptimal solution for problem (P0). Finally, simulation results demonstrate that i) the designed algorithm can converge faster and aggregate model more accurately compared to baselines; ii) the training loss and prediction accuracy of federated learning can be improved significantly with the aid of multiple RISs.
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