# An optimal algorithm for proper learning of unions of two rectangles with queries

## Abstract

We study the problem of proper learning of unions of two discretized axis-parallel rectangles over the domain {0,*n*−1}^{d} in the on-line model with equivalence and membership queries. An obvious approach to this problem would use two equivalence queries to find one example in each of the two rectangles contained in the target concept and then use membership queries to find end points of the rectangles. However, there is one substantial difficulty: For any two end points, how to decide whether they belong to the same rectangle? In this paper, we develop some strategies to overcome the above difficulties and construct an algorithm that properly learns unions of two rectangles over the domain {0,*n*−1}^{d} with at most two equivalence queries and at most (11*d*+2) log *n*+d+3 membership queries. We also show that this algorithm is optimal in terms of query complexity

## Preview

Unable to display preview. Download preview PDF.

## References

- [A]D. Angluin, “Queries and concept learning”,
*Machine Learning*, 2, 1988, pages 319–342.Google Scholar - [Am]F. Ameur, “A space-bounded learning algorithm for axis-parallel rectangles”, EuroCOLT'95.Google Scholar
- [Au]P. Auer, “On-line learning of rectangles in noisy environment”,
*Proc of the 6th Annual Workshop on Computational Learning Theory*, 1993, pages 253–261.Google Scholar - [BEHW]A. Blumer, A. Ehrenfeucht, D. David, and M. Warmuth, “Learnability and the Vapnik-Chervonenkis dimension”,
*J. ACM*, pages 929–965, 1989.Google Scholar - [BCH]N. Bshouty, Z. Chen, S. Homer, “On learning discretized geometric concepts”,
*Proc of the 35th Annual Symposium on Foundations of Computer Science*, pages 54–63, 1994.Google Scholar - [BGGM]N. Bshouty, P. Goldberg, S. Goldman, and D. Mathias, “Exact learning of discretized geometric concepts”, Technical Report WUCS-94-19, Dept of Computer Science, Washington University at St. Louis, July, 1994.Google Scholar
- [BM]W. Bultman, W. Maass, “Fast identification of geometric objects with membership queries”,
*Proc of the 4th Annual ACM Workshop on Computational Learning Theory*, pages 337–353, 1991.Google Scholar - [C]Z. Chen, “Learning unions of two rectangles in the plane with equivalence queries”,
*Proc of the 6th Annual ACM Conference on Computational Learning Theory*, pages 243–253, 1993.Google Scholar - [CHa]Z. Chen, S. Homer, “Learning unions of rectangles with queries”, Technical Report BUCS-93-10, Dept of Computer Science, Boston University, July, 93.Google Scholar
- [CHb]Z. Chen, S. Homer, “The bounded injury priority method and the learnability of unions of rectangles”, accepted to publish in
*Annals of Pure and Applied Logic*.Google Scholar - [CMa]Z. Chen, W. Maass, “On-line learning of rectangles”,
*Proc of the 5th Annual Workshop on Computational Learning Theory*, pages 16–28, 1992.Google Scholar - [CMb]Z. Chen, W. Maass, “On-line learning of rectangles and unions of rectangles”,
*Machine Learning*vol. 17, pages 201–223, 1994.Google Scholar - [GGM]P. Goldberg, S. Goldman, and D. Mathias, “Learning unions of rectangles with membership and equivalence queries”,
*Proc of the 7th annual ACM Conference on Computational Learning Theory*, pages 198–207, 1994.Google Scholar - [J]J. Jackson, “An efficient membership-query algorithm for learning DNF with respect to the uniform distribution”,
*Proc of the 35th Annual Symposium on Foundations of Computer Science*, pages 42–53, 1994.Google Scholar - [L]N. Littlestone, “Learning quickly when irrelevant attributes abound: a new linear threshold algorithm”,
*Machine Learning*, 2, 1987, pages 285–318.Google Scholar - [LW]P. Long, M. Warmuth, “Composite geometric concepts and polynomial predictability”,
*Proc of the 3th Annual Workshop on Computational Learning Theory*, pages 273–287, 1991.Google Scholar - [MTa]W. Maass, G. Turán, “On the complexity of learning from counterexamples”,
*Proc of the 30th Annual Symposium on Foundations of Computer Science*, 1989, pages 262–267.Google Scholar - [MTb]W. Maass, G. Turán, “On the complexity of learning from counterexamples and membership queries”,
*Proc of the 31th Annual Symposium on Foundations of Computer Science*, 1990, pages 203–210.Google Scholar - [MTd]W. Maass, G. Turán, “Algorithms and lower bounds for on-line learning of geometric concepts”,
*Machine Learning*, 1994, pages 251–269.Google Scholar - [MW]W. Maass, M. Warmuth, “Efficient learning with virtual threshold gates”, Technical Report 395 of the Institutes for Information Processing Graz, August, 1994.Google Scholar
- [PRa]K. Pillaipakkamnatt, V. Raghavan, “On the limits of proper learnability of subclasses of DNF formulas”,
*Proc of the 7th annual ACM Conference on Computational Learning Theory*, pages 118–129, 1994.Google Scholar - [PRb]K. Pillaipakkamnatt, V. Raghavan, “Read-twice DNF formulas are properly learnable”, Technical Report TR-93-58, Department of Computer Science, Vanderbilt University, 1993.Google Scholar
- [PV]L. Pitt, L. G. Valiant, “Computational limitations on learning from examples”,
*J. of the ACM*, 35, 1988, 965–984.CrossRefGoogle Scholar - [V]L. Valiant, “A theory of the learnable”,
*Comm. of the ACM*, 27, 1984, pages 1134–1142.CrossRefGoogle Scholar