  Previous theoretical work on deep learning and neural network optimization
tend to focus on avoiding saddle points and local minima. However, the
practical observation is that, at least in the case of the most successful Deep
Convolutional Neural Networks (DCNNs), practitioners can always increase the
network size to fit the training data (an extreme example would be [1]). The
most successful DCNNs such as VGG and ResNets are best used with a degree of
"overparametrization". In this work, we characterize with a mix of theory and
experiments, the landscape of the empirical risk of overparametrized DCNNs. We
first prove in the regression framework the existence of a large number of
degenerate global minimizers with zero empirical error (modulo inconsistent
equations). The argument that relies on the use of Bezout theorem is rigorous
when the RELUs are replaced by a polynomial nonlinearity (which empirically
works as well). As described in our Theory III [2] paper, the same minimizers
are degenerate and thus very likely to be found by SGD that will furthermore
select with higher probability the most robust zero-minimizer. We further
experimentally explored and visualized the landscape of empirical risk of a
DCNN on CIFAR-10 during the entire training process and especially the global
minima. Finally, based on our theoretical and experimental results, we propose
an intuitive model of the landscape of DCNN's empirical loss surface, which
might not be as complicated as people commonly believe.
