On the Eeective Vc Dimension

The very idea of an \EEective Vapnik Chervonenkis (VC) dimension" (Vapnik, Levin and Le Cun, 1993) relies on the hypothesis that the relation between the generalization error and the number of training examples can be expressed by a formula algebraically similar to the VC bound. This hypothesis calls for a serious discussion since the traditional VC bound widely overestimates the generalization error. In this paper we describe an algorithm and data dependent measure of capacity. We derive a conndence interval on the diierence between the training error and the generalization error. This conndence interval is much tighter than the traditional VC bound. A simple change of the formulation of the problem yields this extra accuracy: our conndence interval bounds the error diierence between a training set and a test set, rather than the error diierence between a training set and some hypothetical grand truth. This \transductive" approach allows for deriving a data and algorithm dependent conndence interval.