Distributed Over-the-air Computing for Fast Distributed Optimization: Beamforming Design and Convergence Analysis
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Distributed optimization concerns the optimization of a common function in a distributed network, which finds a wide range of applications ranging from machine learning to vehicle platooning. Its key operation is to aggregate all local state information (LSI) at devices to update their states. The required extensive message exchange and many iterations cause a communication bottleneck when the LSI is high dimensional or at high mobility. To overcome the bottleneck, we propose in this work the framework of distributed over-the-air computing (AirComp) to realize a one-step aggregation for distributed optimization by exploiting simultaneous multicast beamforming of all devices and the property of analog waveform superposition of a multi-access channel. We consider two design criteria. The first 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 second criterion, 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. In this case, the optimal beamforming admits closed form. Both the MMSE and ZF beamforming exhibit a centroid structure resulting from averaging columns of conventional MMSE/ZF precoding. Last, the convergence of a classic distributed optimization algorithm is analyzed. The distributed AirComp is found to accelerate convergence by dramatically reducing communication latency. Another key finding is that the ZF beamforming outperforms the MMSE design as the latter is shown to cause bias in subgradient estimation.
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