Empirical Margin Distributions and Bounding the Generalization Error of Combined Classifiers

New probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifier combinations, based on the methods of the theory of Gaussian and empirical processes are proved.

Abstract

We prove new probabilistic upper bounds on generalization error of\ncomplex classifiers that are combinations of simple classifiers. Such\ncombinations could be implemented by neural networks or by voting methods of\ncombining the classifiers, such as boosting and bagging. The bounds are in\nterms of the empirical distribution of the margin of the combined classifier.\nThey are based on the methods of the theory of Gaussian and empirical processes\n(comparison inequalities, symmetrization method, concentration inequalities)\nand they improve previous results of Bartlett (1998) on bounding the\ngeneralization error of neural networks in terms of $\\ell_1$-norms of the\nweights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding\nthe generalization error of boosting. We also obtain rates of convergence in\nLévy distance of empirical margin distribution to the true margin\ndistribution uniformly over the classes of classifiers and prove the optimality\nof these rates.