  In recent years, stochastic gradient descent (SGD) based techniques has
become the standard tools for training neural networks. However, formal
theoretical understanding of why SGD can train neural networks in practice is
largely missing.
  In this paper, we make progress on understanding this mystery by providing a
convergence analysis for SGD on a rich subset of two-layer feedforward networks
with ReLU activations. This subset is characterized by a special structure
called "identity mapping". We prove that, if input follows from Gaussian
distribution, with standard $O(1/\sqrt{d})$ initialization of the weights, SGD
converges to the global minimum in polynomial number of steps. Unlike normal
vanilla networks, the "identity mapping" makes our network asymmetric and thus
the global minimum is unique. To complement our theory, we are also able to
show experimentally that multi-layer networks with this mapping have better
performance compared with normal vanilla networks.
  Our convergence theorem differs from traditional non-convex optimization
techniques. We show that SGD converges to optimal in "two phases": In phase I,
the gradient points to the wrong direction, however, a potential function $g$
gradually decreases. Then in phase II, SGD enters a nice one point convex
region and converges. We also show that the identity mapping is necessary for
convergence, as it moves the initial point to a better place for optimization.
Experiment verifies our claims.
