On Teaching and Learning Intersection-Closed Concept Classes

Abstract

We consider the self-directed learning model [7] which is a variant of Littlestone’s mistake-bound model [9,10]. We will refute the conjecture of [8,2] that for intersection-closed concept classes, the self-directed learning complexity is related to the VC-dimension. We show that, even under the assumption of intersection-closedness, both parameters are completely incomparable.

We furthermore investigate the structure of intersection-closed concept classes which are difficult to learn in the self-directed learning model. We show that such classes must contain maximum classes.

We consider the teacher-directed learning model [5] in the worst, best and average case performance. While the teaching complexity in the worst case is incomparable to the VC-dimension, large concept classes (e.g. balls) are bounded by VC-dimension with respect to the average case. We show that the teaching complexity in the best case is bounded by the self-directed learning complexity. It is also bounded by the VCdimension, if the concept class is intersection-closed. This does not hold for arbitrary concept classes. We find examples which substantiate this gap.

The author likes to thank Hans Ulrich Simon, Norbert Klasner, Michael Schmitt, Andreas Birkendorf and unidentified referees for helpful hints and discussions. The author gratefully acknowledges the support of the German-Israeli Foundation for Scientific Research and Development (grant I-403-001.06/95).