  In the mean field regime, neural networks are appropriately scaled so that as
the width tends to infinity, the learning dynamics tends to a nonlinear and
nontrivial dynamical limit, known as the mean field limit. This lends a way to
study large-width neural networks via analyzing the mean field limit. Recent
works have successfully applied such analysis to two-layer networks and
provided global convergence guarantees. The extension to multilayer ones
however has been a highly challenging puzzle, and little is known about the
optimization efficiency in the mean field regime when there are more than two
layers.
  In this work, we prove a global convergence result for unregularized
feedforward three-layer networks in the mean field regime. We first develop a
rigorous framework to establish the mean field limit of three-layer networks
under stochastic gradient descent training. To that end, we propose the idea of
a \textit{neuronal embedding}, which comprises of a fixed probability space
that encapsulates neural networks of arbitrary sizes. The identified mean field
limit is then used to prove a global convergence guarantee under suitable
regularity and convergence mode assumptions, which -- unlike previous works on
two-layer networks -- does not rely critically on convexity. Underlying the
result is a universal approximation property, natural of neural networks, which
importantly is shown to hold at \textit{any} finite training time (not
necessarily at convergence) via an algebraic topology argument.
