A New PAC Bound for Intersection-Closed Concept Classes

Abstract

For hyper-rectangles in R ^d Auer et al. [1] proved a PAC bound of $O(\frac{1}{\epsilon}(d+{\rm log}\frac{1}{\delta}))$, where ε and δ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension d in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\epsilon}(d{\rm log}d+\frac{1}{\delta}))$ for arbitrary intersection-closed concept classes complementing the well-known bounds $O(\frac{1}{\epsilon}({\rm log}\frac{1}{\delta}+d{\rm log}\frac{1}{\epsilon}))$ and $O(\frac{d}{\epsilon}{\rm log}\frac{1}{\delta})$ of Blumer et al. and Haussler et al. [4,6]. Our bound is established using the closure algorithm, that generates as its hypothesis the smallest concept that is consistent with the positive training examples. On the other hand, we show that maximum intersection-closed concept classes meet the bound of $O(\frac{1}{\epsilon}(d+{\rm log}\frac{1}{\delta}))$ as well. Moreover, we indicate that our new as well as the conjectured bound cannot hold for arbitrary consistent learning algorithms, giving an example of such an algorithm that needs $\Omega(\frac{1}{\epsilon}(d+{\rm log}\frac{1}{\epsilon}+{\rm log \frac{1}{\delta}}))$ examples to learn some simple maximum intersection-closed concept class.