  It is widely observed that deep learning models with learned parameters
generalize well, even with much more model parameters than the number of
training samples. We systematically investigate the underlying reasons why deep
neural networks often generalize well, and reveal the difference between the
minima (with the same training error) that generalize well and those they
don't. We show that it is the characteristics the landscape of the loss
function that explains the good generalization capability. For the landscape of
loss function for deep networks, the volume of basin of attraction of good
minima dominates over that of poor minima, which guarantees optimization
methods with random initialization to converge to good minima. We theoretically
justify our findings through analyzing 2-layer neural networks; and show that
the low-complexity solutions have a small norm of Hessian matrix with respect
to model parameters. For deeper networks, extensive numerical evidence helps to
support our arguments.
