PAC Learning Axis-aligned Rectangles with Respect toProduct Distributions from Multiple-Instance Examples

Authors:
Philip M. Long;Lei Tan
Affiliations:
ISCS Department, National University of Singapore, Singapore 119260, Republic of Singapore. E-mail: plong@iscs.nus.sg;One Microsoft Way, Redmond, WA 98052. E-mail: raymondt@microsoft.com
Venue:
Machine Learning - Special issue on the ninth annual conference on computational theory (COLT '96)
Year:
1998

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Abstract

We describe a polynomial-time algorithm for learning axis-alignedrectangles in Q^d with respect to product distributionsfrom multiple-instance examples in the PAC model. Here, each exampleconsists of n elements of Q^d together witha label indicating whether any of the n points is in therectangle to be learned. We assume that there is an unknown productdistribution D over Q^d such that allinstances are independently drawn according to D. The accuracyof a hypothesis is measured by the probability that it would incorrectlypredict whether one of n more points drawn from Dwas in the rectangle to be learned. Our algorithm achieves accuracy εwith probability 1-δ in O\left(\frac{d^5n^{12}}{\epsilon^{20}} \log^2 \frac{nd}{\epsilon\delta}\right) time.