  In this paper, we study the implicit regularization of the gradient descent
algorithm in homogeneous neural networks, including fully-connected and
convolutional neural networks with ReLU or LeakyReLU activations. In
particular, we study the gradient descent or gradient flow (i.e., gradient
descent with infinitesimal step size) optimizing the logistic loss or
cross-entropy loss of any homogeneous model (possibly non-smooth), and show
that if the training loss decreases below a certain threshold, then we can
define a smoothed version of the normalized margin which increases over time.
We also formulate a natural constrained optimization problem related to margin
maximization, and prove that both the normalized margin and its smoothed
version converge to the objective value at a KKT point of the optimization
problem. Our results generalize the previous results for logistic regression
with one-layer or multi-layer linear networks, and provide more quantitative
convergence results with weaker assumptions than previous results for
homogeneous smooth neural networks. We conduct several experiments to justify
our theoretical finding on MNIST and CIFAR-10 datasets. Finally, as margin is
closely related to robustness, we discuss potential benefits of training longer
for improving the robustness of the model.
