Combinatorial probability and the tightness of generalization bounds

Accurate prediction of the generalization ability of a learning algorithm is an important problem in computational learning theory. The classical Vapnik-Chervonenkis (VC) generalization bounds are too general and therefore overestimate the expected error. Recently obtained data-dependent bounds are still overestimated. To find out why the bounds are loose, we reject the uniform convergence principle and apply a purely combinatorial approach that is free of any probabilistic assumptions, makes no approximations, and provides an empirical control of looseness. We introduce new data-dependent complexity measures: a local shatter coefficient and a nonscalar local shatter profile, which can give much tighter bounds than the classical VC shatter coefficient. An experiment on real datasets shows that the effective local measures may take very small values; thus, the effective local VC dimension takes values in [0, 1] and therefore is not related to the dimension of the space. © 2008 Pleiades Publishing, Ltd.