Error Bounds for Real Function Classes Based on Discretized Vapnik-Chervonenkis Dimensions

The Vapnik-Chervonenkis (VC) dimension plays an important role in statistical learning theory. In this paper, we propose the discretized VC dimension obtained by discretizing the range of a real function class. Then, we point out that Sauer’s Lemma is valid for the discretized VC dimension. We group the real function classes having the infinite VC dimension into four categories by using the discretized VC dimension. As a byproduct, we present the equidistantly discretized VC dimension by introducing an equidistant partition to segmenting the range of a real function class. Finally, we obtain the error bounds for real function classes based on the discretized VC dimensions in the PAC-learning framework.