Spectrally-normalized margin bounds for neural networks

TL;DR

This paper introduces a new margin-based generalization bound for neural networks that scales with spectral complexity, linking model complexity, training dynamics, and generalization performance.

Contribution

It provides a spectral-normalized margin bound for neural networks and empirically demonstrates its correlation with model complexity and generalization on standard datasets.

Findings

Bound correlates with spectral norms and risk

SGD selects predictors with complexity related to task difficulty

Bound is sensitive to neural network complexity

Abstract

This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.