Learning unions of high-dimensional boxes over the reals

Abstract

Beimel and Kushilevitz (1997) presented an algorithm that exactly learns (using membership queries and equivalence queries) several classes of unions of boxes in high dimension over finite discrete domains. The running time of the algorithm is polynomial in the logarithm of the size of the domain and other parameters of the target function (in particular, the dimension). We go one step further and present a PAC+MQ algorithm whose running time is independent of the size of the domain. Thus, we can learn such classes of boxes over infinite domains. Specifically, we learn unions of t disjoint n-dimensional boxes over the reals in time polynomial in n and t, and unions of O(log n) (possibly intersecting) n-dimensional boxes over the reals in time polynomial in n.