Abstract
We use the notion of general dimension to show that any p-evaluatable concept class with polynomial query complexity can be learned in polynomial time with the help of an oracle in the polynomial hierarchy, where the complexity of the required oracle depends on the query-types used by the learning algorithm. In particular, we show that for subset and superset queries an oracle in ∑p 3 suffices. Since the concept class of DNF formulas has polynomial query complexity with respect to subset and superset queries with DNF formulas as hypotheses, it follows that DNF formulas are properly learnable in polynomial time with subset and superset queries and the help of an oracle in ∑p 3 . We also show that the required oracle in our main theorem cannot be replaced by an oracle in a lower level of the polynomial-time hierarchy, unless the hierarchy collapses.
Work supported by the DFG under project KO 1053/1-1
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Aizenstein, T. Hegedüs, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity, 7:19–53, 1998.
D. Angluin. Queries and concept learning. Machine Learning, 2:319–342, 1988.
D. Angluin and M. Kharitonov. When won’t membership queries help? Journal of Computer and System Sciences, 50:336–355, 1995.
J. Balcázar, J. Castro, and D. Guijarro. Abstract combinatorial characterizations of exact learning via queries. In Proc. 13th ACM Conference on Computational Learning Theory, pages 248–254. Morgan Kaufmann, 2000.
J. Balcázar, J. Castro, and D. Guijarro. A general dimension for exact learning. In Proc. 14th ACM Conference on Computational Learning Theory, volume 2111 of Lecture Notes in Artificial Intelligence, pages 354–367. Springer-Verlag, Berlin Heidelberg New York, 2001.
N. Bshouty, R. Cleve, R. Gavaldà, S. Kannan, and C. Tamon. Oracles and queries that are sufficient for exact learning. Journal of Computer and System Sciences, 52:421–433, 1996.
L. Hellerstein and M. Karpinsky. Learning read-once formulas using membership queries. In Proc. 2nd ACM Conference on Computational Learning Theory, pages 146–161. Morgan Kaufmann, 1989.
L. Hellerstein, K. Pillaipakkamnatt, V. Raghavan, and D. Wilkins. How many queries are needed to learn? Journal of the ACM, 43(5):840–862, 1996.
M. J. Kearns and L. G. Valiant. Cryptographic limitations on learning boolean formulae and finite automata. Journal of the ACM, 41:67–95, 1994.
J. Köbler. Lowness-Eigenschaften und Erlernbarkeit von Booleschen Schaltkreisklassen. Habilitationsschrift, Universität Ulm, 1995.
J. Köbler and W. Lindner. Oracles in ∑p/2 are sufficient for exact learning. International Journal of Foundations of Computer Science, 11(4):615–632, 2000.
C. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
O. Watanabe. A framework for polynomial time query learnability. Mathematical Systems Theory, 27:211–229, 1994.
O. Watanabe and R. Gavaldà. Structural analysis of polynomial time query learnability. Mathematical Systems Theory, 27:231–256, 1994.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Köbler, J., Lindner, W. (2002). The Complexity of Learning Concept Classes with Polynomial General Dimension. In: Cesa-Bianchi, N., Numao, M., Reischuk, R. (eds) Algorithmic Learning Theory. ALT 2002. Lecture Notes in Computer Science(), vol 2533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36169-3_14
Download citation
DOI: https://doi.org/10.1007/3-540-36169-3_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00170-6
Online ISBN: 978-3-540-36169-5
eBook Packages: Springer Book Archive