  Decentralized federated learning (DFL), inherited from distributed
optimization, is an emerging paradigm to leverage the explosively growing data
from wireless devices in a fully distributed manner.DFL enables joint training
of machine learning model under device to device (D2D) communication fashion
without the coordination of a parameter server. However, the deployment of
wireless DFL is facing some pivotal challenges. Communication is a critical
bottleneck due to the required extensive message exchange between neighbor
devices to share the learned model. Besides, consensus becomes increasingly
difficult as the number of devices grows because there is no available central
server to perform coordination. To overcome these difficulties, this paper
proposes employing over-the-air computation (Aircomp) to improve communication
efficiency by exploiting the superposition property of analog waveform in
multi-access channels, and introduce the mixing matrix mechanism to promote
consensus using the spectral property of symmetric doubly stochastic matrix.
Specifically, we develop a novel multiple-input multiple-output over-the-air
DFL (MIMO OA-DFL) framework to study over-the-air DFL problem over MIMO
multiple access channels. We conduct a general convergence analysis to
quantitatively capture the influence of aggregation weight and communication
error on the MIMO OA-DFL performance in \emph{ad hoc} networks. The result
shows that the communication error together with the spectral gap of mixing
matrix has a significant impact on the learning performance. Based on this, a
joint communication-learning optimization problem is formulated to optimize
transceiver beamformers and mixing matrix. Extensive numerical experiments are
performed to reveal the characteristics of different topologies and demonstrate
the substantial learning performance enhancement of our proposed algorithm.
