  This note explores the applicability of unsupervised machine learning
techniques towards hard optimization problems on random inputs. In particular
we consider Graph Neural Networks (GNNs) -- a class of neural networks designed
to learn functions on graphs -- and we apply them to the max-cut problem on
random regular graphs. We focus on the max-cut problem on random regular graphs
because it is a fundamental problem that has been widely studied. In
particular, even though there is no known explicit solution to compare the
output of our algorithm to, we can leverage the known asymptotics of the
optimal max-cut value in order to evaluate the performance of the GNNs.
  In order to put the performance of the GNNs in context, we compare it with
the classical semidefinite relaxation approach by Goemans and Williamson~(SDP),
and with extremal optimization, which is a local optimization heuristic from
the statistical physics literature. The numerical results we obtain indicate
that, surprisingly, Graph Neural Networks attain comparable performance to the
Goemans and Williamson SDP. We also observe that extremal optimization
consistently outperforms the other two methods. Furthermore, the performances
of the three methods present similar patterns, that is, for sparser, and for
larger graphs, the size of the found cuts are closer to the asymptotic optimal
max-cut value.
