Abstract
We study the problem of proper learning of unions of two discretized axis-parallel rectangles over the domain {0,nā1}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,nā1}d with at most two equivalence queries and at most (11d+2) log n+d+3 membership queries. We also show that this algorithm is optimal in terms of query complexity
The author was supported by NSF grants CCR-9103055 and CCR-9400229.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
D. Angluin, āQueries and concept learningā, Machine Learning, 2, 1988, pages 319ā342.
F. Ameur, āA space-bounded learning algorithm for axis-parallel rectanglesā, EuroCOLT'95.
P. Auer, āOn-line learning of rectangles in noisy environmentā, Proc of the 6th Annual Workshop on Computational Learning Theory, 1993, pages 253ā261.
A. Blumer, A. Ehrenfeucht, D. David, and M. Warmuth, āLearnability and the Vapnik-Chervonenkis dimensionā, J. ACM, pages 929ā965, 1989.
N. Bshouty, Z. Chen, S. Homer, āOn learning discretized geometric conceptsā, Proc of the 35th Annual Symposium on Foundations of Computer Science, pages 54ā63, 1994.
N. Bshouty, P. Goldberg, S. Goldman, and D. Mathias, āExact learning of discretized geometric conceptsā, Technical Report WUCS-94-19, Dept of Computer Science, Washington University at St. Louis, July, 1994.
W. Bultman, W. Maass, āFast identification of geometric objects with membership queriesā, Proc of the 4th Annual ACM Workshop on Computational Learning Theory, pages 337ā353, 1991.
Z. Chen, āLearning unions of two rectangles in the plane with equivalence queriesā, Proc of the 6th Annual ACM Conference on Computational Learning Theory, pages 243ā253, 1993.
Z. Chen, S. Homer, āLearning unions of rectangles with queriesā, Technical Report BUCS-93-10, Dept of Computer Science, Boston University, July, 93.
Z. Chen, S. Homer, āThe bounded injury priority method and the learnability of unions of rectanglesā, accepted to publish in Annals of Pure and Applied Logic.
Z. Chen, W. Maass, āOn-line learning of rectanglesā, Proc of the 5th Annual Workshop on Computational Learning Theory, pages 16ā28, 1992.
Z. Chen, W. Maass, āOn-line learning of rectangles and unions of rectanglesā, Machine Learning vol. 17, pages 201ā223, 1994.
P. Goldberg, S. Goldman, and D. Mathias, āLearning unions of rectangles with membership and equivalence queriesā, Proc of the 7th annual ACM Conference on Computational Learning Theory, pages 198ā207, 1994.
J. Jackson, āAn efficient membership-query algorithm for learning DNF with respect to the uniform distributionā, Proc of the 35th Annual Symposium on Foundations of Computer Science, pages 42ā53, 1994.
N. Littlestone, āLearning quickly when irrelevant attributes abound: a new linear threshold algorithmā, Machine Learning, 2, 1987, pages 285ā318.
P. Long, M. Warmuth, āComposite geometric concepts and polynomial predictabilityā, Proc of the 3th Annual Workshop on Computational Learning Theory, pages 273ā287, 1991.
W. Maass, G. TurĆ”n, āOn the complexity of learning from counterexamplesā, Proc of the 30th Annual Symposium on Foundations of Computer Science, 1989, pages 262ā267.
W. Maass, G. TurĆ”n, āOn the complexity of learning from counterexamples and membership queriesā, Proc of the 31th Annual Symposium on Foundations of Computer Science, 1990, pages 203ā210.
W. Maass, G. TurĆ”n, āAlgorithms and lower bounds for on-line learning of geometric conceptsā, Machine Learning, 1994, pages 251ā269.
W. Maass, M. Warmuth, āEfficient learning with virtual threshold gatesā, Technical Report 395 of the Institutes for Information Processing Graz, August, 1994.
K. Pillaipakkamnatt, V. Raghavan, āOn the limits of proper learnability of subclasses of DNF formulasā, Proc of the 7th annual ACM Conference on Computational Learning Theory, pages 118ā129, 1994.
K. Pillaipakkamnatt, V. Raghavan, āRead-twice DNF formulas are properly learnableā, Technical Report TR-93-58, Department of Computer Science, Vanderbilt University, 1993.
L. Pitt, L. G. Valiant, āComputational limitations on learning from examplesā, J. of the ACM, 35, 1988, 965ā984.
L. Valiant, āA theory of the learnableā, Comm. of the ACM, 27, 1984, pages 1134ā1142.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
Ā© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, Z. (1995). An optimal algorithm for proper learning of unions of two rectangles with queries. In: Du, DZ., Li, M. (eds) Computing and Combinatorics. COCOON 1995. Lecture Notes in Computer Science, vol 959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0030848
Download citation
DOI: https://doi.org/10.1007/BFb0030848
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60216-3
Online ISBN: 978-3-540-44733-7
eBook Packages: Springer Book Archive