  Wide networks are often believed to have a nice optimization landscape, but
what rigorous results can we prove? To understand the benefit of width, it is
important to identify the difference between wide and narrow networks. In this
work, we prove that from narrow to wide networks, there is a phase transition
from having sub-optimal basins to no sub-optimal basins. Specifically, we prove
two results: on the positive side, for any continuous activation functions, the
loss surface of a class of wide networks has no sub-optimal basins, where
"basin" is defined as the set-wise strict local minimum; on the negative side,
for a large class of networks with width below a threshold, we construct strict
local minima that are not global. These two results together show the phase
transition from narrow to wide networks.
