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Distributed optimization concerns the optimization of a common function in adistributed network, which finds a wide range of applications ranging frommachine learning to vehicle platooning. Its key operation is to aggregate alllocal state information (LSI) at devices to update their states. The requiredextensive message exchange and many iterations cause a communication bottleneckwhen the LSI is high dimensional or at high mobility. To overcome thebottleneck, we propose in this work the framework of distributed over-the-aircomputing (AirComp) to realize a one-step aggregation for distributedoptimization by exploiting simultaneous multicast beamforming of all devicesand the property of analog waveform superposition of a multi-access channel. Weconsider two design criteria. The first one is to minimize the sum AirComperror (i.e., sum mean-squared error (MSE)) with respect to the desiredaverage-functional values. An efficient solution approach is proposed bytransforming the non-convex beamforming problem into an equivalentconcave-convex fractional program and solving it by nesting convex programminginto a bisection search. The second criterion, called zero-forcing (ZF)multicast beamforming, is to force the received over-the-air aggregated signalsat devices to be equal to the desired functional values. In this case, theoptimal beamforming admits closed form. Both the MMSE and ZF beamformingexhibit a centroid structure resulting from averaging columns of conventionalMMSE/ZF precoding. Last, the convergence of a classic distributed optimizationalgorithm is analyzed. The distributed AirComp is found to accelerateconvergence by dramatically reducing communication latency. Another key findingis that the ZF beamforming outperforms the MMSE design as the latter is shownto cause bias in subgradient estimation.

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