Training binary perceptron by clipping

Abstract

A teacher perceptron T with N binary components provides the classification of a set of p randomly chosen training examples. Several algorithms are available that use this information to select a student perceptron J with continuous components Ji. The purpose is to maximize the overlap R = T·J/(|T| |J|), or to minimize the corresponding generalization error ε = (1/π) arccos R. In view of the binary nature of the components of the teacher, one might expect that a lower error can be achieved by working with the clipped version of the student vector, namely the vector with components sign (Ji). It turns out that this is not always the case. In this letter we calculate the overlap for a vector with components f(Ji), where f can be any odd function of its argument, as a function of the overlap R. We show that the optimal choice of f is a hyperbolic tangent f(x) = th ((R/(1 - R2)x)). The corresponding generalization error can go to zero exponentially fast in a2, for a large (a = p/N).