Abstract
Recent work has introduced Boolean kernels with which one can learn over a feature space containing all conjunctions of length up to k (for any 1≤ k ≤ n) over the original n Boolean features in the input space. This motivates the question of whether maximum margin algorithms such as support vector machines can learn Disjunctive Normal Form expressions in the PAC learning model using this kernel. We study this question, as well as a variant in which structural risk minimization (SRM) is performed where the class hierarchy is taken over the length of conjunctions.
We show that such maximum margin algorithms do not PAC learn t(n)-term DNF for any t(n) = ω(1), even when used with such a SRM scheme. We also consider PAC learning under the uniform distribution and show that if the kernel uses conjunctions of length \(\tilde{\omega}(\sqrt{n})\) then the maximum margin hypothesis will fail on the uniform distribution as well. Our results concretely illustrate that margin based algorithms may overfit when learning simple target functions with natural kernels.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Blum, A., Furst, M., Jackson, J., Kearns, M., Mansour, Y., Rudich, S.: Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In: Proceedings of the 26th Annual Symposium on Theory of Computing, pp. 253–262 (1994)
Blum, A., Rudich, S.: Fast learning of k-term DNF formulas with queries. Journal of Computer and System Sciences 51(3), 367–373 (1995)
Boser, B., Guyon, I., Vapnik, V.: A training algorithm for optimal margin classifiers. In: Proceedings of the 5th Annual Workshop on Computational Learning Theory, pp. 144–152 (1992)
Bshouty, N.: A subexponential exact learning algorithm for DNF using equivalence queries. Information Processing Letters 59, 37–39 (1996)
Bshouty, N., Tamon, C.: On the Fourier spectrum of monotone functions. Journal of the ACM 43(4), 747–770 (1996)
Gentile, C.: A new approximate maximal margin classification algorithm. Journal of Machine Learning Research 2, 213–242 (2001)
Hancock, T., Mansour, Y.: Learning monotone k-μ DNF formulas on product distributions. In: Proceedings of the 4th Annual Conference on Computational Learning Theory, pp. 179–193 (1991)
Jackson, J.: An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences 55, 414–440 (1997)
Kearns, M., Vazirani, U.: An introduction to computational learning theory. MIT Press, Cambridge (1994)
Khardon, R.: On using the Fourier transform to learn disjoint DNF. Information Processing Letters 49, 219–222 (1994)
Khardon, R., Roth, D., Servedio, R.: Efficiency versus convergence of boolean kernels for on-line learning algorithms. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems, Cambridge, MA, vol. 14. MIT Press, Cambridge (2002)
Klivans, A., Servedio, R.: Learning DNF in time 2õ(n 1/3). In: Proceedings of the Thirty-Third Annual Symposium on Theory of Computing, pp. 258–265 (2001)
Kowalczyk, A., Smola, A.J., Williamson, R.C.: Kernel machines and boolean functions. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems, Cambridge, MA, vol. 14. MIT Press, Cambridge (2002)
Kucera, L., Marchetti-Spaccamela, A., Protassi, M.: On learning monotone DNF formulae under uniform distributions. Information and Computation 110, 84–95 (1994)
Kushilevitz, E., Roth, D.: On learning visual concepts and DNF formulae. In: Proceedings of the 6th Annual Conference on Computational Learning Theory, pp. 317–326 (1993)
Minsky, M., Papert, S.: Perceptrons: an introduction to computational geometry. MIT Press, Cambridge (1968)
Sadohara, K.: Learning of boolean functions using support vector machines. In: Abe, N., Khardon, R., Zeugmann, T. (eds.) ALT 2001. LNCS (LNAI), vol. 2225, pp. 106–118. Springer, Heidelberg (2001)
Sakai, Y., Maruoka, A.: Learning monotone log-term DNF formulas under the uniform distribution. Theory of Computing Systems 33, 17–33 (2000)
Servedio, R.: On PAC learning using winnow, perceptron, and a perceptron-like algorithm. In: Proceedings of the 12th Annual Conference on Computational Learning Theory, pp. 296–307 (1999)
Servedio, R.: On learning monotone DNF under product distributions. In: Proceedings of the Fourteenth Annual Conference on Computational Learning Theory, pp. 473–489 (2001)
Shawe-Taylor, J., Cristianini, N.: An introduction to support vector machines. Cambridge University Press, Cambridge (2000)
Tarui, J., Tsukiji, T.: Learning DNF by approximating inclusion-exclusion formulae. In: Proceedings of the Fourteenth Conference on Computational Complexity, pp. 215–220 (1999)
Valiant, L.: A theory of the learnable. Communications of the ACM 27(11), 1134–1142 (1984)
Verbeurgt, K.: Learning DNF under the uniform distribution in quasi-polynomial time. In: Proceedings of the Third Annual Workshop on Computational Learning Theory, pp. 314–326 (1990)
Verbeurgt, K.: Learning sub-classes of monotone DNF on the uniform distribution. In: Proceedings of the 9th Conference on Algorithmic Learning Theory, pp. 385–399 (1998)
Watkins, C.: Kernels from matching operations. Technical Report CSD-TR-98-07, Computer Science Department, Royal Holloway, University of London (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Khardon, R., Servedio, R.A. (2003). Maximum Margin Algorithms with Boolean Kernels. In: Schölkopf, B., Warmuth, M.K. (eds) Learning Theory and Kernel Machines. Lecture Notes in Computer Science(), vol 2777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45167-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-45167-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40720-1
Online ISBN: 978-3-540-45167-9
eBook Packages: Springer Book Archive