  Implementing Decentralized Gradient Descent (DGD) in wireless systems is
challenging due to noise, fading, and limited bandwidth, necessitating topology
awareness, transmission scheduling, and the acquisition of channel state
information (CSI) to mitigate interference and maintain reliable
communications. These operations may result in substantial signaling overhead
and scalability challenges in large networks lacking central coordination. This
paper introduces a scalable DGD algorithm that eliminates the need for
scheduling, topology information, or CSI (both average and instantaneous). At
its core is a Non-Coherent Over-The-Air (NCOTA) consensus scheme that exploits
a noisy energy superposition property of wireless channels. Nodes encode their
local optimization signals into energy levels within an OFDM frame and transmit
simultaneously, without coordination. The key insight is that the received
energy equals, on average, the sum of the energies of the transmitted signals,
scaled by their respective average channel gains, akin to a consensus step.
This property enables unbiased consensus estimation, utilizing average channel
gains as mixing weights, thereby removing the need for their explicit design or
for CSI. Introducing a consensus stepsize mitigates consensus estimation errors
due to energy fluctuations around their expected values. For strongly-convex
problems, it is shown that the expected squared distance between the local and
globally optimum models vanishes at a rate of $\mathcal O(1/\sqrt{k})$ after
$k$ iterations, with suitable decreasing learning and consensus stepsizes.
Extensions accommodate a broad class of fading models and frequency-selective
channels. Numerical experiments on image classification demonstrate faster
convergence in terms of running time compared to state-of-the-art schemes,
especially in dense network scenarios.
