  The loss surfaces of deep neural networks have been the subject of several
studies, theoretical and experimental, over the last few years. One strand of
work considers the complexity, in the sense of local optima, of high
dimensional random functions with the aim of informing how local optimisation
methods may perform in such complicated settings. Prior work of Choromanska et
al (2015) established a direct link between the training loss surfaces of deep
multi-layer perceptron networks and spherical multi-spin glass models under
some very strong assumptions on the network and its data. In this work, we test
the validity of this approach by removing the undesirable restriction to ReLU
activation functions. In doing so, we chart a new path through the spin glass
complexity calculations using supersymmetric methods in Random Matrix Theory
which may prove useful in other contexts. Our results shed new light on both
the strengths and the weaknesses of spin glass models in this context.
