Asymptotic law of likelihood ratio for multilayer perceptron models

TL;DR

This paper derives the exact asymptotic distribution of likelihood ratio statistics for multilayer perceptron models, especially when the number of hidden units is over-estimated, revealing a supremum of Gaussian process squares.

Contribution

It provides the first detailed asymptotic law of likelihood ratio for over-estimated MLP models, addressing a gap in statistical theory for neural networks.

Findings

Likelihood ratio statistic converges to the supremum of Gaussian process squares.

Asymptotic distribution differs from classical chi-squared law in over-estimated models.

Results apply when model parameters are in a compact set.

Abstract

We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The data are assumed to be generated by a true MLP model and the estimation of the parameters of the MLP is done by maximizing the likelihood of the model. When the number of hidden units of the true model is known, the asymptotic distribution of the maximum likelihood estimator (MLE) and the likelihood ratio (LR) statistic is easy to compute and converge to a $\chi^2$ law. However, if the number of hidden unit is over-estimated the Fischer information matrix of the model is singular and the asymptotic behavior of the MLE is unknown. This paper deals with this case, and gives the exact asymptotic law of the LR statistics. Namely, if the parameters of the MLP lie in a suitable compact set, we show that the LR statistics is the supremum of the square of a Gaussian process indexed by a class of limit score functions.