  Finding parameters in a deep neural network (NN) that fit training data is a
nonconvex optimization problem, but a basic first-order optimization method
(gradient descent) finds a global optimizer with perfect fit (zero-loss) in
many practical situations. We examine this phenomenon for the case of Residual
Neural Networks (ResNet) with smooth activation functions in a limiting regime
in which both the number of layers (depth) and the number of weights in each
layer (width) go to infinity. First, we use a mean-field-limit argument to
prove that the gradient descent for parameter training becomes a gradient flow
for a probability distribution that is characterized by a partial differential
equation (PDE) in the large-NN limit. Next, we show that under certain
assumptions, the solution to the PDE converges in the training time to a
zero-loss solution. Together, these results suggest that the training of the
ResNet gives a near-zero loss if the ResNet is large enough. We give estimates
of the depth and width needed to reduce the loss below a given threshold, with
high probability.
