  Neural networks, a central tool in machine learning, have demonstrated
remarkable, high fidelity performance on image recognition and classification
tasks. These successes evince an ability to accurately represent high
dimensional functions, but rigorous results about the approximation error of
neural networks after training are few. Here we establish conditions for global
convergence of the standard optimization algorithm used in machine learning
applications, stochastic gradient descent (SGD), and quantify the scaling of
its error with the size of the network. This is done by reinterpreting SGD as
the evolution of a particle system with interactions governed by a potential
related to the objective or "loss" function used to train the network. We show
that, when the number $n$ of units is large, the empirical distribution of the
particles descends on a convex landscape towards the global minimum at a rate
independent of $n$, with a resulting approximation error that universally
scales as $O(n^{-1})$. These properties are established in the form of a Law of
Large Numbers and a Central Limit Theorem for the empirical distribution. Our
analysis also quantifies the scale and nature of the noise introduced by SGD
and provides guidelines for the step size and batch size to use when training a
neural network. We illustrate our findings on examples in which we train neural
networks to learn the energy function of the continuous 3-spin model on the
sphere. The approximation error scales as our analysis predicts in as high a
dimension as $d=25$.
