Learning boxes in high dimension

We present exact learning algorithms that learn several classes of (discrete) boxes in {0, ..., ℓ - 1}ⁿ. In particular we learn: (1) The class of unions of O(log n) boxes in time poly(n, log ℓ) (solving an open problem of [16] and [12]; in [3] this class is shown to be learnable in time poly(n, ℓ)). (2) The class of unions of disjoint boxes in time poly(n, t, log ℓ), where t is the number of boxes. (Previously this was known only in the case where all boxes are disjoint in one of the dimensions; in [3] this class is shown to be learnable in time poly(n, t, ℓ).) In particular our algorithm learns the class of decision trees over n variables, that take values in {0,..., ℓ - 1}, with comparison nodes in time poly(n, t, log ℓ), where t is the number of leaves (this was an open problem in [9] which was shown in [4] to be learnable in time poly(n, t, ℓ)). (3) The class of unions of O(1)-degenerate boxes (that is, boxes that depend only on O(1) variables) in time poly(n, t, log ℓ) (generalizing the learnability of O(1)-DNF and of boxes in O(1) dimensions). The algorithm for this class uses only equivalence queries and it can also be used to learn the class of unions of O(1) boxes (from equivalence queries only).