  In this paper, we present some theoretical work to explain why simple
gradient descent methods are so successful in solving non-convex optimization
problems in learning large-scale neural networks (NN). After introducing a
mathematical tool called canonical space, we have proved that the objective
functions in learning NNs are convex in the canonical model space. We further
elucidate that the gradients between the original NN model space and the
canonical space are related by a pointwise linear transformation, which is
represented by the so-called disparity matrix. Furthermore, we have proved that
gradient descent methods surely converge to a global minimum of zero loss
provided that the disparity matrices maintain full rank. If this full-rank
condition holds, the learning of NNs behaves in the same way as normal convex
optimization. At last, we have shown that the chance to have singular disparity
matrices is extremely slim in large NNs. In particular, when over-parameterized
NNs are randomly initialized, the gradient decent algorithms converge to a
global minimum of zero loss in probability.
