Regularization learning, early stopping and biased estimator

In this article, we present a unified statistical interpretation of regularization learning and early stopping for linear networks in the context of statistical regression; i.e. linear regression model. Firstly, those concepts are shown to be equivalent with the use of a biased estimator under the purpose of constructing the network with lower generalization error than the least-squares estimator. It is also found that the biased estimator is a shrinkage estimator. Secondly, we showed that the optimal regularization parameter or the optimal stopping time according to the generalization error are obtained by solving the bias/variance dilemma. Lastly, we gave the estimates of the optimal regularization parameter and the optimal stopping time based on the training data. Simple numerical simulations showed that those estimates are possible to improve the generalization error compared with the least-squares estimator. Additionally, we discussed the relationship between the Bayesian interpretation of the regularization parameter and the optimal regularization parameter which minimizes the generalization error. (C) 2002 Elsevier Science B.V. All rights reserved.