We present a PAC-learning algorithm and an on-line learning algorithm for nested differences of intersection-closed classes. Examples of intersection-closed classes include axis-parallel rectangles, monomials, and linear sub-spaces. Our PAC-learning algorithm uses a pruning technique that we rigorously proof correct. As a result we show that the tolerable noise rate for this algorithm does not depend on the complexity (VC-dimension) of the target class but only on the VC-dimension of the underlying intersection-closed class. For our on-line algorithm we show an optimal mistake bound in the sense that there are concept classes for which each on-line learning algorithm (using nested differences as hypotheses) can be forced to make at least that many mistakes.
Auer, P. (1997). Learning nested differences in the presence of malicious noise. Theoretical Computer Science, 185(1), 159–175. https://doi.org/10.1016/S0304-3975(97)00019-4