A Spectral Approach to Generalization and Optimization in Neural Networks

The recent success of deep neural networks stems from their ability to generalize well on real data; however, Zhang et al. have observed that neural networks can easily overfit random labels. This observation demonstrates that with the existing theory, we cannot adequately explain why gradient methods can find generalizable solutions for neural networks. In this work, we use a Fourier-based approach to study the generalization properties of gradient-based methods over 2-layer neural networks with sinusoidal activation functions. We prove that if the underlying distribution of data has nice spectral properties such as bandlimitedness, then the gradient descent method will converge to generalizable local minima. We also establish a Fourier-based generalization bound for bandlimited spaces, which generalizes to other activation functions. Our generalization bound motivates a grouped version of path norms for measuring the complexity of 2-layer neural networks with ReLU activation functions. We demonstrate numerically that regularization of this group path norm results in neural network solutions that can fit true labels without losing test accuracy while not overfitting random labels.