  The key to generalization is controlling the complexity of the network.
However, there is no obvious control of complexity -- such as an explicit
regularization term -- in the training of deep networks for classification. We
will show that a classical form of norm control -- but kind of hidden -- is
present in deep networks trained with gradient descent techniques on
exponential-type losses. In particular, gradient descent induces a dynamics of
the normalized weights which converge for $t \to \infty$ to an equilibrium
which corresponds to a minimum norm (or maximum margin) solution. For
sufficiently large but finite $\rho$ -- and thus finite $t$ -- the dynamics
converges to one of several margin maximizers, with the margin monotonically
increasing towards a limit stationary point of the flow. In the usual case of
stochastic gradient descent, most of the stationary points are likely to be
convex minima corresponding to a constrained minimizer -- the network with
normalized weights-- which corresponds to vanishing regularization. The
solution has zero generalization gap, for fixed architecture, asymptotically
for $N \to \infty$, where $N$ is the number of training examples. Our approach
extends some of the original results of Srebro from linear networks to deep
networks and provides a new perspective on the implicit bias of gradient
descent. We believe that the elusive complexity control we describe is
responsible for the puzzling empirical finding of good predictive performance
by deep networks, despite overparametrization.
