Rational activation functions in neural networks with uniform based loss functions and its application in classification

Abstract: In this paper, we demonstrate the application of generalised rational uniform (Chebyshev) approximation in neural networks. In particular, our activation functions are one degree rational functions and the loss function is based on the uniform norm. In this setting, when the coefficients of the rational activation function are fixed, the overall optimisation problem of the neural network forms a generalised rational uniform approximation problem where the weights and the bias of the network are the decision variables. To optimise the decision variables, we suggest using two prominent methods: the bisection method and the differential correction algorithm. We perform numerical experiments on classification problems with two classes and report the classification accuracy obtained by the network using the bisection method, differential correction algorithm along with the standard MATLAB toolbox which uses the least square loss function. We show that the choice of the uniform norm based loss function with rational activation function and the bisection method lead to better classification accuracy when the training dataset is either very small or if the classes are imbalanced.