  Machine learning, and in particular neural network models, have
revolutionized fields such as image, text, and speech recognition. Today, many
important real-world applications in these areas are driven by neural networks.
There are also growing applications in engineering, robotics, medicine, and
finance. Despite their immense success in practice, there is limited
mathematical understanding of neural networks. This paper illustrates how
neural networks can be studied via stochastic analysis, and develops approaches
for addressing some of the technical challenges which arise. We analyze
one-layer neural networks in the asymptotic regime of simultaneously (A) large
network sizes and (B) large numbers of stochastic gradient descent training
iterations. We rigorously prove that the empirical distribution of the neural
network parameters converges to the solution of a nonlinear partial
differential equation. This result can be considered a law of large numbers for
neural networks. In addition, a consequence of our analysis is that the trained
parameters of the neural network asymptotically become independent, a property
which is commonly called "propagation of chaos".
