# A new PAC bound for intersection-closed concept classes

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## Abstract

For hyper-rectangles in \(\mathbb{R}^{d}\) Auer (1997) proved a PAC bound of \(O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))\), where \(\varepsilon\) and \(\delta\) 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 \(O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))\) for arbitrary intersection-closed concept classes, complementing the well-known bounds \(O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))\) and \(O(\frac{d}{\varepsilon}\log \frac{1}{\delta})\) of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the *closure algorithm*, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of \(O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))\). For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need \(\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))\) examples to learn some particular maximum intersection-closed concept classes.

### Keywords

PAC bounds Intersection-closed classes### References

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