  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.
