  Over-parameterized neural networks generalize well in practice without any
explicit regularization. Although it has not been proven yet, empirical
evidence suggests that implicit regularization plays a crucial role in deep
learning and prevents the network from overfitting. In this work, we introduce
the gradient gap deviation and the gradient deflection as statistical measures
corresponding to the network curvature and the Hessian matrix to analyze
variations of network derivatives with respect to input parameters, and
investigate how implicit regularization works in ReLU neural networks from both
theoretical and empirical perspectives. Our result reveals that the network
output between each pair of input samples is properly controlled by random
initialization and stochastic gradient descent to keep interpolating between
samples almost straight, which results in low complexity of over-parameterized
neural networks.
