Training with Noise is Equivalent to Tikhonov Regularization

TLDR

Adding noise to neural network inputs during training is known to improve generalization, and prior work has shown this is equivalent to adding a regularization term to the error function, though that term involves unbounded second derivatives and can cause difficulties. In this paper we show that for network training the regularization term can be reduced to a positive semi‑definite form involving only first derivatives of the network mapping. For a sum‑of‑squares loss, the regularization term is a generalized Tikhonov regularizer, and its direct minimization offers a practical alternative to training with noise.

Abstract

It is well known that the addition of noise to the input data of a neural network during training can, in some circumstances, lead to significant improvements in generalization performance. Previous work has shown that such training with noise is equivalent to a form of regularization in which an extra term is added to the error function. However, the regularization term, which involves second derivatives of the error function, is not bounded below, and so can lead to difficulties if used directly in a learning algorithm based on error minimization. In this paper we show that for the purposes of network training, the regularization term can be reduced to a positive semi-definite form that involves only first derivatives of the network mapping. For a sum-of-squares error function, the regularization term belongs to the class of generalized Tikhonov regularizers. Direct minimization of the regularized error function provides a practical alternative to training with noise.