Distributed Over-the-Air Computing for Fast Distributed Optimization: Beamforming Design and Convergence Analysis

Distributed optimization finds a wide range of applications ranging from machine learning to vehicle platooning. To overcome the bottleneck caused by the required extensive message exchange, we propose in this work the framework of distributed over-the-air computing (AirComp) to realize a one-step aggregation for distributed optimization. Equivalently, the technique superimposes multiple instances of conventional AirComp processes, giving rise to the challenge of jointly designing multicast beamforming at devices to rein in errors due to interference and channel distortion. We consider two design criteria. One is to minimize the sum AirComp error (i.e., sum mean-squared error (MSE)) with respect to the desired average-functional values. An efficient solution approach is proposed by transforming the non-convex beamforming problem into an equivalent concave-convex fractional program and solving it by nesting convex programming into a bisection search. The other one, called zero-forcing (ZF) multicast beamforming, is to force the received over-the-air aggregated signals at devices to be equal to the desired functional values, where the optimal beamforming admits closed form. Last, the convergence of a classic distributed optimization algorithm is analyzed. The distributed AirComp is found experimentally to accelerate convergence by dramatically reducing communication latency.