A New PAC Bound for Intersection-Closed Concept Classes

For hyper-rectangles in Rd Auer et al. [1] proved a PAC bound of O 1 (d + log 1 ) , where and 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 1 (d log d + log 1 ) for arbitrary intersection-closed concept classes, complementing the well-known bounds O 1 (log 1 + d log 1 ) and O d log 1 of Blumer et al. [4] and Haussler et al. [7]. 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 intersectionclosed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of O 1 (d + log 1 ) . For such improved bounds the c...

• Type: Conference
• Year: 2004
• Where: COLT
• Authors: Peter Auer, Ronald Ortner
• Keywords: COLT 2004, D Log, Intersection-closed Concept Classes, Machine Learning, PAC Bound