The problem of designing cost functions to estimate a posteriori probabilities in multiclass problems is addressed in this paper. We establish necessary and sufficient conditions that these costs must satisfy in one-class one-output networks whose outputs are consistent with probability laws. We focus our attention on a particular subset of the corresponding cost functions; those which verify two usually interesting properties: symmetry and separability (well-known cost functions, such as the quadratic cost or the cross entropy are particular cases in this subset). Finally, we present a universal stochastic gradient learning rule for single-layer networks, in the sense of minimizing a general version of these cost functions for a wide family of nonlinear activation functions.

}, keywords = {Cost functions, Estimation, Functions, Learning algorithms, Multiclass problems, Neural networks, Pattern recognition, Probability, Problem solving, Random processes, Stochastic gradient learning rule}, issn = {10459227}, doi = {10.1109/72.761724}, url = {http://www.scopus.com/inward/record.url?eid=2-s2.0-0032643080\&partnerID=40\&md5=d528195bd6ec84531e59ddd2ececcd46}, author = {Jes{\'u}s Cid-Sueiro and Juan I. Arribas and S Urban-Munoz and A R Figueiras-Vidal} }