  Federated learning is a privacy-preserving approach to train a global model
at a central server by collaborating with wireless devices, each with its own
local training data set. In this paper, we present a compressive sensing
approach for federated learning over massive multiple-input multiple-output
communication systems in which the central server equipped with a massive
antenna array communicates with the wireless devices. One major challenge in
system design is to reconstruct local gradient vectors accurately at the
central server, which are computed-and-sent from the wireless devices. To
overcome this challenge, we first establish a transmission strategy to
construct sparse transmitted signals from the local gradient vectors at the
devices. We then propose a compressive sensing algorithm enabling the server to
iteratively find the linear minimum-mean-square-error (LMMSE) estimate of the
transmitted signal by exploiting its sparsity. We also derive an analytical
threshold for the residual error at each iteration, to design the stopping
criterion of the proposed algorithm. We show that for a sparse transmitted
signal, the proposed algorithm requires less computationally complexity than
LMMSE. Simulation results demonstrate that the presented approach outperforms
conventional linear beamforming approaches and reduces the performance gap
between federated learning and centralized learning with perfect
reconstruction.
