title: Are loss functions all the same?
abstract: In this letter, we investigate the impact of choosing different loss functions from the viewpoint of statistical learning theory. We introduce a convexity assumption, which is met by all loss functions commonly used in the literature, and study how the bound on the estimation error changes with the loss. We also derive a general result on the minimizer of the expected risk for a convex loss function in the case of classification. The main outcome of our analysis is that for classification, the hinge loss appears to be the loss of choice. Other things being equal, the hinge loss leads to a convergence rate practically indistinguishable from the logistic loss rate and much better than the square loss rate. Furthermore, if the hypothesis space is sufficiently rich, the bounds obtained for the hinge loss are not loosened by the thresholding stage.
authors.affiliation: INFM-DISI, Università di Genova, 16146 Genoa, Italy. rosasco@disi.unige.it
authors.forename: Lorenzo
authors.initials: L
authors.lastname: Rosasco
authors.forename: Ernesto
authors.initials: E
authors.lastname: De Vito
authors.forename: Andrea
authors.initials: A
authors.lastname: Caponnetto
authors.forename: Michele
authors.initials: M
authors.lastname: Piana
authors.forename: Alessandro
authors.initials: A
authors.lastname: Verri
mesh_terms: D007858:Learning / Q000502:physiology; D016014:Linear Models; D008959:Models, Neurological; D013223:Statistics as Topic