  We consider a broadband over-the-air computation empowered model aggregation
approach for wireless federated learning (FL) systems and propose to leverage
an intelligent reflecting surface (IRS) to combat wireless fading and noise. We
first investigate the conventional node-selection based framework, where a few
edge nodes are dropped in model aggregation to control the aggregation error.
We analyze the performance of this node-selection based framework and derive an
upper bound on its performance loss, which is shown to be related to the
selected edge nodes. Then, we seek to minimize the mean-squared error (MSE)
between the desired global gradient parameters and the actually received ones
by optimizing the selected edge nodes, their transmit equalization
coefficients, the IRS phase shifts, and the receive factors of the cloud
server. By resorting to the matrix lifting technique and difference-of-convex
programming, we successfully transform the formulated optimization problem into
a convex one and solve it using off-the-shelf solvers. To improve learning
performance, we further propose a weight-selection based FL framework. In such
a framework, we assign each edge node a proper weight coefficient in model
aggregation instead of discarding any of them to reduce the aggregation error,
i.e., amplitude alignment of the received local gradient parameters from
different edge nodes is not required. We also analyze the performance of this
weight-selection based framework and derive an upper bound on its performance
loss, followed by minimizing the MSE via optimizing the weight coefficients of
the edge nodes, their transmit equalization coefficients, the IRS phase shifts,
and the receive factors of the cloud server. Furthermore, we use the MNIST
dataset for simulations to evaluate the performance of both node-selection and
weight-selection based FL frameworks.
