A weighted hybrid training algorithm of neural networks for robust data regression

Both the hybrid full memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm (termed as HFM) and its extension with regularisation learning framework (RHFM) have been widely used in training feed-forward neural networks, where the main feature lies in deploying an alternative optimisation (AO) technique for the hidden parameter learning (based on gradient-based optimisation) and the output weights evaluation (according to linear least square problem). However, either HFM or RHFM can perform effectively and favourably in data modelling with training samples contaminated by noises or outliers. This paper develops a robust learning algorithm by weighting the residual error resulted from RHFM, named by WRHFM, aiming to control the contribution of each training sample for the regularised cost function. For algorithm implementation, the AO technique is employed in updating iteratively the weighting factors with well-defined statistical estimates, the hidden parameters with gradient descending, and the output weights with a closed form solution. Performance evaluation is carried out on a function approximation example and several real-world data sets, where comparisons against error back-propagation, support vector machine (SVM), HFM and RHFM has demonstrated the advantages of the proposed WRHFM algorithm in data regression with outliers.