  For distributed learning among collaborative users, this paper develops and
analyzes a communication-efficient scheme for federated learning (FL) over the
air, which incorporates 1-bit compressive sensing (CS) into analog aggregation
transmissions. To facilitate design parameter optimization, we theoretically
analyze the efficacy of the proposed scheme by deriving a closed-form
expression for the expected convergence rate of the FL over the air. Our
theoretical results reveal the tradeoff between convergence performance and
communication efficiency as a result of the aggregation errors caused by
sparsification, dimension reduction, quantization, signal reconstruction and
noise. Then, we formulate 1-bit CS based FL over the air as a joint
optimization problem to mitigate the impact of these aggregation errors through
joint optimal design of worker scheduling and power scaling policy. An
enumeration-based method is proposed to solve this non-convex problem, which is
optimal but becomes computationally infeasible as the number of devices
increases. For scalable computing, we resort to the alternating direction
method of multipliers (ADMM) technique to develop an efficient implementation
that is suitable for large-scale networks. Simulation results show that our
proposed 1-bit CS based FL over the air achieves comparable performance to the
ideal case where conventional FL without compression and quantification is
applied over error-free aggregation, at much reduced communication overhead and
transmission latency.
