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

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 e with probability 1-δ in O\left(\frac{d^5n^{12}}{\epsilon^{20}} \log^2 \frac{nd}{\epsilon\delta}\right) time.