  The aim of this work is to develop a fully-distributed algorithmic framework
for training graph convolutional networks (GCNs). The proposed method is able
to exploit the meaningful relational structure of the input data, which are
collected by a set of agents that communicate over a sparse network topology.
After formulating the centralized GCN training problem, we first show how to
make inference in a distributed scenario where the underlying data graph is
split among different agents. Then, we propose a distributed gradient descent
procedure to solve the GCN training problem. The resulting model distributes
computation along three lines: during inference, during back-propagation, and
during optimization. Convergence to stationary solutions of the GCN training
problem is also established under mild conditions. Finally, we propose an
optimization criterion to design the communication topology between agents in
order to match with the graph describing data relationships. A wide set of
numerical results validate our proposal. To the best of our knowledge, this is
the first work combining graph convolutional neural networks with distributed
optimization.
