Optimization in an Error Backpropagation Neural Network Environment with a Performance Test on a Spectral Pattern Classification Problem

This paper attempts to develop a mathematically rigid framework for minimising the cross-entropy function in an error backpropagating framework. In doing so, we derive the backpropagation formulae for evaluating the partial derivatives in a computationally efficient way. Various techniques of optimising the multiple-class cross-entropy error function to train single hidden layer neural network classifiers with softmax output transfer functions are investigated on a real world multispectral pixel-by-pixel classification problem that is of fundamental importance in remote sensing. These techniques include epoch-based and batch versions of backpropagation of gradient descent, PR-conjugate gradient, and BFGS quasi-Newton errors. The method of choice depends upon the nature of the learning task and whether one wants to optimise learning for speed or classification performance. It was found that, comparatively considered, gradient descent error backpropagation provided the best and most stable out-of-sample performance results across batch and epoch-based modes of operation. If the goal is to maximise learning speed and a sacrifice in classification accuracy is acceptable, then PR-conjugate gradient error backpropagation tends to be superior. If the training set is very large, stochastic epoch-based versions of local optimisers should be chosen utilising a larger rather than a smaller epoch size to avoid unacceptable instabilities in the classification results.