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Federated learning (FL) with over-the-air computation can efficiently utilizethe communication bandwidth but is susceptible to analog aggregation error.Excluding those devices with weak channel conditions can reduce the aggregationerror, but it also limits the amount of local training data for FL, which canreduce the training convergence rate. In this work, we jointly design uplinkreceiver beamforming and device selection for over-the-air FL over time-varyingwireless channels to maximize the training convergence rate. We reformulatethis stochastic optimization problem into a mixed-integer program using anupper bound on the global training loss over communication rounds. We thenpropose a Greedy Spatial Device Selection (GSDS) approach, which uses asequential procedure to select devices based on a measure capturing both thechannel strength and the channel correlation to the selected devices. We showthat given the selected devices, the receiver beamforming optimization problemis equivalent to downlink single-group multicast beamforming. To reduce thecomputational complexity, we also propose an Alternating-optimization-basedDevice Selection and Beamforming (ADSBF) approach, which solves the receiverbeamforming and device selection subproblems alternatingly. In particular,despite the device selection being an integer problem, we are able to developan efficient algorithm to find its optimal solution.  Simulation results with real-world image classification demonstrate that ourproposed methods achieve faster convergence with significantly lowercomputational complexity than existing alternatives. Furthermore, althoughADSBF shows marginally inferior performance to GSDS, it offers the advantage oflower computational complexity when the number of devices is large.

Beamforming and Device Selection Design in Federated Learning with Over-the-air Aggregation