Is Pocket algorithm optimal?

Most learning algorithms for single neuron are not able to provide for any classification problem the weight vector which satisfies the maximum number of input-output relations contained in the training set. An important exception is given by the pocket algorithm: it repeatedly executes the perceptron algorithm and maintains (in the pocket) the weight vector which is remained unchanged for the highest number of iterations. A proper convergence theorem ensures the achievement of an optimal configuration with probability one when the number of iterations grows indefinitely. This theoretical result is used for showing the good convergence properties of other learning methods for multilayered neural networks, which employ pocket algorithm as a basic task. In the present paper a new formulation of the pocket convergence theorem is given; a rigorous proof corrects some formal and substantial errors which invalidate previous theoretical results. In particular it is shown that the optimality of the asymptotical solution is ensured only if the number of permanences for the pocket vector lies in a proper interval of the real axis which bounds depend on the number of iterations.