  This theoretical paper is devoted to developing a rigorous theory for
demystifying the global convergence phenomenon in a challenging scenario:
learning over-parameterized Rectified Linear Unit (ReLU) nets for very high
dimensional dataset under very mild assumptions. A major ingredient of our
analysis is a fine-grained analysis of random activation matrices. The
essential virtue of dissecting activation matrices is that it bridges the
dynamics of optimization and angular distribution in high-dimensional data
space. This angle-based detailed analysis leads to asymptotic characterizations
of gradient norm and directional curvature of objective function at each
gradient descent iteration, revealing that the empirical loss function enjoys
nice geometrical properties in the overparameterized setting. Along the way, we
significantly improve existing theoretical bounds on both over-parameterization
condition and learning rate with very mild assumptions for learning very high
dimensional data. Moreover, we uncover the role of the geometrical and spectral
properties of the input data in determining desired over-parameterization size
and global convergence rate. All these clues allow us to discover a novel
geometric picture of nonconvex optimization in deep learning: angular
distribution in high-dimensional data space $\mapsto$ spectrums of
overparameterized activation matrices $\mapsto$ favorable geometrical
properties of empirical loss landscape $\mapsto$ global convergence phenomenon.
Furthremore, our theoretical results imply that gradient-based nonconvex
optimization algorithms have much stronger statistical guarantees with much
milder over-parameterization condition than exisiting theory states for
learning very high dimensional data, which is rarely explored so far.
