# PAC Learning Axis-aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples

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## Abstract

We describe a polynomial-time algorithm for learning axis-aligned rectangles in Q^{ d } with respect to product distributions from multiple-instance examples in the PAC model. Here, each example consists of *n* elements of Q^{d} together with a label indicating whether any of the *n* points is in the rectangle to be learned. We assume that there is an unknown product distribution *D* over Q^{ d } such that all instances are independently drawn according to *D*. The accuracy of a hypothesis is measured by the probability that it would incorrectly predict whether one of *n* more points drawn from *D* was in the rectangle to be learned. Our algorithm achieves accuracy ∈ with probability *1-*δ in O (d^{5} n^{12}/∈^{20} log^{2} nd/∈δ time.

PAC learning multiple-instance examples axis-aligned hyperrectangles

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© Kluwer Academic Publishers 1998