  Over-the-air computation (AirComp) based federated learning (FL) is capable
of achieving fast model aggregation by exploiting the waveform superposition
property of multiple access channels. However, the model aggregation
performance is severely limited by the unfavorable wireless propagation
channels. In this paper, we propose to leverage intelligent reflecting surface
(IRS) to achieve fast yet reliable model aggregation for AirComp-based FL. To
optimize the learning performance, we formulate an optimization problem that
jointly optimizes the device selection, the aggregation beamformer at the base
station (BS), and the phase shifts at the IRS to maximize the number of devices
participating in the model aggregation of each communication round under
certain mean-squared-error (MSE) requirements. To tackle the formulated
highly-intractable problem, we propose a two-step optimization framework.
Specifically, we induce the sparsity of device selection in the first step,
followed by solving a series of MSE minimization problems to find the maximum
feasible device set in the second step. We then propose an alternating
optimization framework, supported by the difference-of-convex-functions
programming algorithm for low-rank optimization, to efficiently design the
aggregation beamformers at the BS and phase shifts at the IRS. Simulation
results will demonstrate that our proposed algorithm and the deployment of an
IRS can achieve a lower training loss and higher FL prediction accuracy than
the baseline algorithms.
