Canonical ensemble approach to graded-response perceptrons

Perceptrons with graded input-output relations and a limited output precision are studied within the Gardner-Derrida canonical ensemble approach. Soft non-negative error measures are introduced allowing for extended retrieval properties. In particular, the performance of these systems for a linear (quadratic) error measure, corresponding to the perceptron (adaline) learning algorithm, is compared with the performance for a rigid error measure, simply counting the number of errors. Replica-symmetry-breaking effects are evaluated, and the analytic results are compared with numerical simulations.