A jamming transition from under- to over-parametrization affects generalization in deep learning

In this paper we first recall the recent result that in deep networks a phase transition, analogous to the jamming transition of granular media, delimits the over- and under-parametrized regimes where fitting can or cannot be achieved. The analysis leading to this result support that for proper initialization and architectures, in the whole over-parametrized regime poor minima of the loss are not encountered during training, because the number of constraints that hinders the dynamics is insufficient to allow for the emergence of stable minima. Next, we study systematically how this transition affects generalization properties of the network (i.e. its predictive power). As we increase the number of parameters of a given model, starting from an under-parametrized network, we observe for gradient descent that the generalization error displays three phases: (i) initial decay, (ii) increase until the transition point?where it displays a cusp?and (iii) slow decay toward an asymptote as the network width diverges. However if early stopping is used, the cusp signaling the jamming transition disappears. Thereby we identify the region where the classical phenomenon of over-fitting takes place as the vicinity of the jamming transition, and the region where the model keeps improving with increasing the number of parameters, thus organizing previous empirical observations made in modern neural networks.