# Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness

## TL;DR

This paper develops a theoretical framework to quantify deep neural network generalization error based on data complexity and network smoothness, validated through experiments on image datasets.

## Contribution

It introduces the cover complexity measure and inverse modulus of continuity to analyze neural network generalization, linking theoretical bounds with empirical observations.

## Key findings

- Expected error scales with the square root of the number of classes.
- Test loss correlates with neural network smoothness during training.
- Network size affects smoothness, but dataset size does not.

## Abstract

The accuracy of deep learning, i.e., deep neural networks, can be characterized by dividing the total error into three main types: approximation error, optimization error, and generalization error. Whereas there are some satisfactory answers to the problems of approximation and optimization, much less is known about the theory of generalization. Most existing theoretical works for generalization fail to explain the performance of neural networks in practice. To derive a meaningful bound, we study the generalization error of neural networks for classification problems in terms of data distribution and neural network smoothness. We introduce the cover complexity (CC) to measure the difficulty of learning a data set and the inverse of the modulus of continuity to quantify neural network smoothness. A quantitative bound for expected accuracy/error is derived by considering both the CC and neural network smoothness. Although most of the analysis is general and not specific to neural networks, we validate our theoretical assumptions and results numerically for neural networks by several data sets of images. The numerical results confirm that the expected error of trained networks scaled with the square root of the number of classes has a linear relationship with respect to the CC. We also observe a clear consistency between test loss and neural network smoothness during the training process. In addition, we demonstrate empirically that the neural network smoothness decreases when the network size increases whereas the smoothness is insensitive to training dataset size.