  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.
