  The behavior of the gradient descent (GD) algorithm is analyzed for a deep
neural network model with skip-connections. It is proved that in the
over-parametrized regime, for a suitable initialization, with high probability
GD can find a global minimum exponentially fast. Generalization error estimates
along the GD path are also established. As a consequence, it is shown that when
the target function is in the reproducing kernel Hilbert space (RKHS) with a
kernel defined by the initialization, there exist generalizable early-stopping
solutions along the GD path. In addition, it is also shown that the GD path is
uniformly close to the functions given by the related random feature model.
Consequently, in this "implicit regularization" setting, the deep neural
network model deteriorates to a random feature model. Our results hold for
neural networks of any width larger than the input dimension.
