Backpropagation Separates Where Perceptrons Do1

Feedforward nets with sigmoidal activation functions are often designed by minimizing a cost criterion. It has been pointed out before that this technique may be outperformed by the classical perceptron learning rule, at least on some problems. In this paper, we show that no such pathologies can arise if the error criterion is of a threshold LMS type, i.e., is zero for values “beyond” the desired target values. More precisely, we show that if the data are linearly separable, and one considers nets with no hidden neurons, then an error function as above cannot have any local minima that are not global. Simulations of networks with hidden units are consistent with these results, in that often data which can be classified when minimizing a threshold LMS criterion may fail to be classified when using instead a simple LMS cost. In addition, the proof gives the following stronger result, under the stated hypotheses: the continuous gradient adjustment procedure is such that from any initial weight configuration a separating set of weights is obtained in finite time. This is a precise analogue of the Perceptron Learning Theorem. The results are then compared with the more classical pattern recognition problem of threshold LMS with linear activations, where no spurious local minima exist even for nonseparable data: here it is shown that even if using the threshold criterion, such bad local minima may occur, if the data are not separable and sigmoids are used.