Stability Analysis of the Modified Levenberg–Marquardt Algorithm for the Artificial Neural Network Training

Abstract

The Levenberg-Marquardt and Newton are two algorithms that use the Hessian for the artificial neural network learning. In this article, we propose a modified Levenberg-Marquardt algorithm for the artificial neural network learning containing the training and testing stages. The modified Levenberg-Marquardt algorithm is based on the Levenberg-Marquardt and Newton algorithms but with the following two differences to assure the error stability and weights boundedness: 1) there is a singularity point in the learning rates of the Levenberg-Marquardt and Newton algorithms, while there is not a singularity point in the learning rate of the modified Levenberg-Marquardt algorithm and 2) the Levenberg-Marquardt and Newton algorithms have three different learning rates, while the modified Levenberg-Marquardt algorithm only has one learning rate. The error stability and weights boundedness of the modified Levenberg-Marquardt algorithm are assured based on the Lyapunov technique. We compare the artificial neural network learning with the modified Levenberg-Marquardt, Levenberg-Marquardt, Newton, and stable gradient algorithms for the learning of the electric and brain signals data set.