A Fourier-Based Approach to Generalization and Optimization in Deep Learning

Abstract

The success of deep neural networks stems from their ability to generalize well on real data; however, et al. have observed that neural networks can easily overfit randomly-generated labels. This observation highlights the following question: why do gradient methods succeed in finding generalizable solutions for neural networks while there exist solutions with poor generalization behavior? In this work, we use a Fourier-based approach to study the generalization properties of gradient-based methods over 2-layer neural networks with band-limited activation functions. Our results indicate that in such settings if the underlying distribution of data enjoys nice Fourier properties including bandlimitedness and bounded Fourier norm, then the gradient descent method can converge to local minima with nice generalization behavior. We also establish a Fourier-based generalization error bound for band-limited function spaces, applicable to 2-layer neural networks with general activation functions. This generalization bound motivates a grouped version of path norms for measuring the complexity of 2-layer neural networks with ReLU-type activation functions. We empirically demonstrate that regularization of the group path norms results in neural network solutions that can fit true labels without losing test accuracy while not overfitting random labels.