Convergence of online gradient methods for continuous perceptrons with linearly separable training patterns

h this paper, we prove that the online gradient method for continuous perceptrons converges in finite steps when the training patterns are linearly separable. @ 2003 Elsevier Ltd. All rights reserved. Neural networks have been widely used for solving supervised classification problems. In this paper, we consider the simplest feedforword neural network-the perceptron made up of m input neurons and one output neuron. The objective of training the neural networks is, for a given activation function g(x) : R1 + R1, to determine a weight vector W E R " , such that the training patterns {@}jJ=r are correctly classified according to the output C = g(W. f$) (cf. (2)). Some algorithms training the discrete perceptron where g(x) = sgn(x), such as the perceptron rule [l] and the delta rule (or Widrow-Hoff rule [2]) based on the LMS (least mean square) algorithm, have proved convergent for linearly separable training patterns. We are concerned in this paper with the continuous perceptron where g(x) is a sigmoidal function (a continuous function approximating the sign function sgn(x)). In th is case, the online gradient methods are often used for the network training, of which the convergence is our goal in this paper. We expect our analysis here can help to build up similar theories for.more important BP neural networks with hidden layers. In this respect, Gori and Maggini [3] prove a convergence result for BP neural networks with linearly separable patterns, under the assumption that the weight vectors keep bounded in the training process. We do not need this restriction in our case. To train the feedforward neural network (the perceptron), we are supplied with a set of training pattern pairs ([j, Oj}& c R " x {fl}, w h ere the ideal output Oj is " 1 " for a class, and "-1 "