Dropout in Training Neural Networks: Flatness of Solution and Noise Structure

TL;DR

This paper investigates how dropout regularization leads neural networks to flatter minima with noise structures aligned to the Hessian, enhancing understanding of its role in improving generalization across various architectures and datasets.

Contribution

The work provides empirical evidence and theoretical analysis showing dropout induces flatter minima with noise aligned to the Hessian, explaining its effectiveness in regularization.

Findings

Dropout results in flatter minima compared to standard training.

Dropout noise variance is larger in sharper loss landscape directions.

Hessian and dropout noise covariance are similar, explaining dropout's effectiveness.

Abstract

It is important to understand how the popular regularization method dropout helps the neural network training find a good generalization solution. In this work, we show that the training with dropout finds the neural network with a flatter minimum compared with standard gradient descent training. We further find that the variance of a noise induced by the dropout is larger at the sharper direction of the loss landscape and the Hessian of the loss landscape at the found minima aligns with the noise covariance matrix by experiments on various datasets, i.e., MNIST, CIFAR-10, CIFAR-100 and Multi30k, and various structures, i.e., fully-connected networks, large residual convolutional networks and transformer. For networks with piece-wise linear activation function and the dropout is only at the last hidden layer, we then theoretically derive the Hessian and the covariance of dropout randomness, where these two quantities are very similar. This similarity may be a key reason accounting for the goodness of dropout.