Abstract
Predicting class probabilities and other real-valued quantities is often more useful than binary classification, but comparatively little work in PAC-style learning addresses this issue. We show that two rich classes of real-valued functions are learnable in the probabilistic-concept framework of Kearns and Schapire.
Let X be a subset of Euclidean space and f be a real-valued function on X. We say f is a nested halfspace function if, for each real threshold t, the set { x ∈ X | f(x) ≤ t}, is a halfspace. This broad class of functions includes binary halfspaces with a margin (e.g., SVMs) as a special case. We give an efficient algorithm that provably learns (Lipschitz-continuous) nested halfspace functions on the unit ball. The sample complexity is independent of the number of dimensions.
We also introduce the class of uphill decision trees, which are real-valued decision trees (sometimes called regression trees) in which the sequence of leaf values is non-decreasing. We give an efficient algorithm for provably learning uphill decision trees whose sample complexity is polynomial in the number of dimensions but independent of the size of the tree (which may be exponential). Both of our algorithms employ a real-valued extension of Mansour and McAllester’s boosting algorithm.
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
Blum, A., Frieze, A., Kannan, R., Vempala, S.: A polynomial time algorithm for learning noisy linear threshold functions. Algorithmica 22(1/2), 35–52 (1997)
Hastie, T.J., Tibshirani, R.J.: Generalized Additive Models, London: Chapman and Hall (1990)
Kalai, A.: Learning Monotonic Linear Functions. Lecture Notes in Computer Science: Proceedings of the 17th Annual Conference on Learning Theory 3120, 487–501 (2004)
Kearns, M., Mansour, Y.: On the boosting ability of top-down decision tree learning algorithms. Journal of Computer and System Sciences 58, 109–128 (1999)
Kearns, M., Schapire, R.: Efficient distribution-free learning of probabilistic concepts. Journal of Computer and Systems Sciences 48, 464–497 (1994)
Kearns, M., Valiant, L.: Learning boolean formulae or finite automata is as hard as factoring. Technical Report TR-14-88, Harvard University Aiken Computation Laboratory (1988)
Mansour, Y., McAllester, D.: Boosting using branching programs. Journal of Computer and System Sciences 64, 103–112 (2002)
McCullagh, P., Nelder, J.: Generalized Linear Models, Chapman and Hall, London (1989)
McDiarmid, C.: On the method of bounded differences. In: J Siemons, (ed.), Surveys in Combinatorics. London Math Society ( 1989)
O’Donnell, R., Servedio, R.: Learning Monotone Decision Trees in Polynomial Time. In: Proceedings of the 21st Annual Conference on Computational Complexity (CCC), pp. 213–225 (2006)
Schapire, R.: The strength of weak learnability. Machine Learning 5, 197–227 (1990)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Kalai, A.T. (2007). Learning Nested Halfspaces and Uphill Decision Trees. In: Bshouty, N.H., Gentile, C. (eds) Learning Theory. COLT 2007. Lecture Notes in Computer Science(), vol 4539. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72927-3_28
Download citation
DOI: https://doi.org/10.1007/978-3-540-72927-3_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72925-9
Online ISBN: 978-3-540-72927-3
eBook Packages: Computer ScienceComputer Science (R0)