Convex error functions for multilayered perceptrons

For multilayered perceptrons, the error is generally defined as a quadratic function of the difference between the ideal and real outputs. This L² error, considered as a function of the weights, is generally not convex. The author studies the convexity properties of more general error functions for perceptrons with 0 or more hidden layers. A necessary and sufficient condition is presented for the convexity of the error function for all perceptrons without a hidden layer, and an explicit solution is given. A negative result for the convexity for perceptrons is included with one or more than one hidden layers. It is shown how the loss of global convexity for general perceptrons can advantageously be solved in practical applications by the use of a fitted minimization algorithm. An application to the problem of recognition of printed characters is included