  Stochastic gradient descent (SGD) has been found to be surprisingly effective
in training a variety of deep neural networks. However, there is still a lack
of understanding on how and why SGD can train these complex networks towards a
global minimum. In this study, we establish the convergence of SGD to a global
minimum for nonconvex optimization problems that are commonly encountered in
neural network training. Our argument exploits the following two important
properties: 1) the training loss can achieve zero value (approximately), which
has been widely observed in deep learning; 2) SGD follows a star-convex path,
which is verified by various experiments in this paper. In such a context, our
analysis shows that SGD, although has long been considered as a randomized
algorithm, converges in an intrinsically deterministic manner to a global
minimum.
