Theory IIIb: Generalization in Deep Networks
This work addresses a foundational theoretical problem in deep learning by extending margin-based generalization results to nonlinear networks, offering insights for researchers in machine learning theory.
The paper tackles the puzzle of why deep neural networks do not overfit despite high capacity, proving that for multilayer networks with exponential-type losses, gradient descent converges to minimum norm solutions in the separable case, which provides a criterion for ranking generalization performance.
A main puzzle of deep neural networks (DNNs) revolves around the apparent absence of "overfitting", defined in this paper as follows: the expected error does not get worse when increasing the number of neurons or of iterations of gradient descent. This is surprising because of the large capacity demonstrated by DNNs to fit randomly labeled data and the absence of explicit regularization. Recent results by Srebro et al. provide a satisfying solution of the puzzle for linear networks used in binary classification. They prove that minimization of loss functions such as the logistic, the cross-entropy and the exp-loss yields asymptotic, "slow" convergence to the maximum margin solution for linearly separable datasets, independently of the initial conditions. Here we prove a similar result for nonlinear multilayer DNNs near zero minima of the empirical loss. The result holds for exponential-type losses but not for the square loss. In particular, we prove that the weight matrix at each layer of a deep network converges to a minimum norm solution up to a scale factor (in the separable case). Our analysis of the dynamical system corresponding to gradient descent of a multilayer network suggests a simple criterion for ranking the generalization performance of different zero minimizers of the empirical loss.