  It is currently known how to characterize functions that neural networks can
learn with SGD for two extremal parameterizations: neural networks in the
linear regime, and neural networks with no structural constraints. However, for
the main parametrization of interest (non-linear but regular networks) no tight
characterization has yet been achieved, despite significant developments.
  We take a step in this direction by considering depth-2 neural networks
trained by SGD in the mean-field regime. We consider functions on binary inputs
that depend on a latent low-dimensional subspace (i.e., small number of
coordinates). This regime is of interest since it is poorly understood how
neural networks routinely tackle high-dimensional datasets and adapt to latent
low-dimensional structure without suffering from the curse of dimensionality.
Accordingly, we study SGD-learnability with $O(d)$ sample complexity in a large
ambient dimension $d$.
  Our main results characterize a hierarchical property, the "merged-staircase
property", that is both necessary and nearly sufficient for learning in this
setting.
  We further show that non-linear training is necessary: for this class of
functions, linear methods on any feature map (e.g., the NTK) are not capable of
learning efficiently. The key tools are a new "dimension-free" dynamics
approximation result that applies to functions defined on a latent space of
low-dimension, a proof of global convergence based on polynomial identity
testing, and an improvement of lower bounds against linear methods for
non-almost orthogonal functions.
