Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation

TL;DR

This paper introduces data-dependent Rademacher complexity bounds for neural networks that depend on layer norms and Jacobian norms, leading to polynomial depth scaling and improved generalization when regularized during training.

Contribution

It develops new data-dependent complexity bounds considering layer and Jacobian norms, and proposes Jacobian regularization to enhance neural network generalization.

Findings

Bounds scale polynomially with depth when empirical norms are small.

Regularizing Jacobians during training improves test performance.

Theoretical tools for Lipschitz augmentation of function sequences are introduced.

Abstract

Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We suspect that this conundrum comes from the fact that these bounds depend on the training data only through the margin. In practice, many data-dependent techniques such as Batchnorm improve the generalization performance. For feedforward neural nets as well as RNNs, we obtain tighter Rademacher complexity bounds by considering additional data-dependent properties of the network: the norms of the hidden layers of the network, and the norms of the Jacobians of each layer with respect to all previous layers. Our bounds scale polynomially in depth when these empirical quantities are small, as is usually the case in practice. To obtain these bounds, we develop general tools for augmenting a sequence of functions to make their composition Lipschitz and then covering the augmented functions. Inspired by our theory, we directly regularize the network's Jacobians during training and empirically demonstrate that this improves test performance.