Delta rule and learning for min-max neural networks

Abstract

There have been a lot of works discussing (V, Λ)-neural network (see references). However, because of the difficulty of mathematical analysis for (V, Λ)-functions, most previous works choose bounded-plus (+) and multiply (*) as the operations of V and Λ. The (V, Λ) neural network with operators (+, *) is much easier than the (V, Λ) neural network with some other operators, e.g. min-max operators, because it has only a little difference from Back-propagation Neural Network. In this paper, we choose min and max as the operations of V and Λ. Because of the difficulty of functions involved with min and max operations, it is much difficult to deal with (V, Λ) neural network with operators (min, max). In Section 1 of this paper, we first discuss the differentiations of (V, Λ)-functions, and get that 'if f1(x), f2(x), ..., fn(x) are continuously differentiable in real number line R, then any function h(x) generated from f1(x), f2(x), ..., fn(x) through finite times of (V, Λ) operations is continuously differentiable almost everywhere in R'. This statement guarantee that the Delta Rule given in Section 2 is rational and effective. In Section 3 we implement a simple example to show that the Delta Rule given in Section 2 is capable to train (V, Λ) neural networks.