Generalization ability of a perceptron with nonmonotonic transfer function

Abstract

We investigate the generalization ability of a perceptron with nonmonotonic transfer function of a reversed-wedge type in on-line mode. This network is identical to a parity machine, a multilayer network. We consider several learning algorithms. By the perceptron algorithm the generalization error is shown to decrease by the ${\ensuremath{\alpha}}^{\ensuremath{-}1/3}$-law similarly to the case of a simple perceptron in a restricted range of the parameter $a$ characterizing the nonmonotonic transfer function. For other values of $a$, the perceptron algorithm leads to the state where the weight vector of the student is just opposite to that of the teacher. The Hebbian learning algorithm has a similar property; it works only in a limited range of the parameter. The conventional AdaTron algorithm does not give a vanishing generalization error for any values of $a$. We thus introduce a modified AdaTron algorithm that yields a good performance for all values of $a$. We also investigate the effects of optimization of the learning rate as well as of the learning algorithm. Both methods give excellent learning curves proportional to ${\ensuremath{\alpha}}^{\ensuremath{-}1}$. The latter optimization is related to the Bayes statistics and is shown to yield useful hints to extract maximum amount of information necessary to accelerate learning processes.