Advertisement

Robust Inference of Relevant Attributes

  • Jan Arpe
  • Rüdiger Reischuk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2842)

Abstract

Given n Boolean input variables representing a set of attritubes, we consider Boolean functions f (i.e., binary classifications of tuples) that actually depend only on a small but unknown subset of these variables/attributes, in the following called relevant. The goal is to determine the relevant attributes given a sequence of examples – input vectors X and corresponding classifications f(X). We analyze two simple greedy strategies and prove that they are able to achieve this goal for various kinds of Boolean functions and various input distributions according to which the examples are drawn at random.

This generalizes results obtained by Akutsu, Miyano, and Kuhara for the uniform distribution. The analysis also provides explicit upper bounds on the number of necessary examples. They depend on the distribution and combinatorial properties of the function to be inferred.

Our second contribution is an extension of these results to the situation where attribute noise is present, i.e., a certain number of input bits x i may be wrong. This is a typical situation, e.g., in medical research or computational biology, where not all attributes can be measured reliably. We show that even in such an error-prone situation, reliable inference of the relevant attributes can be performed, because our greedy strategies are robust even against a linear number of errors.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrawal, R., Imielinski, T., Swami, A.: Mining Association Rules between Sets of Items in Large Databases. In: Proc. 1993 ACM SIGMOD Conf., pp. 207–216 (1993)Google Scholar
  2. 2.
    Akutsu, T., Bao, F.: Approximating Minimum Keys and Optimal Substructure Screens. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 290–299. Springer, Heidelberg (1996)Google Scholar
  3. 3.
    Akutsu, T., Miyano, S., Kuhara, S.: A Simple Greedy Algorithm for Finding Functional Relations: Efficient Implementation and Average Case Analysis. TCS 292(2), 481–495 (2003); Morishita, S., Arikawa, S. (eds.): DS 2000. LNCS (LNAI), vol. 1967, pp. 86–98. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Angluin, D.: Queries and Concept Learning. Machine Learning 2(4), 319–342 (1988)Google Scholar
  5. 5.
    Angluin, D., Laird, P.: Learning from noisy examples. Machine Learning 2(4), 343–370 (1988)Google Scholar
  6. 6.
    Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations. J. CSS 54, 317–331 (1997)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Arpe, J., Reischuk, R.: Robust Inference of Relevant Attributes. Techn. Report, SIIM-TR-A 03-12, Univ. Lübeck (2003), available at http://www.tcs.mu-luebeck.de/TechReports.html
  8. 8.
    Blum, A., Hellerstein, L., Littlestone, N.: Learning in the Presence of Finitely or Infinitely Many Irrelevant Attributes. In: Proc. 4th, pp. 157–166 (1991)Google Scholar
  9. 9.
    Blum, A., Langley, P.: Selection of Relevant Features and Examples in Machine Learning. Artificial Intelligence 97(1–2), 245–271 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feige, U.: A Threshold of ln n for Approximating Set Cover. J. ACM 45, 634–652 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Goldman, S., Sloan, H.: Can PAC Learning Algorithms Tolerate Random Attribute Noise? Algorithmica 14, 70–84 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Johnson, D.: Approximation Algorithms for Combinatorial Problems. J. CSS 9, 256–278 (1974)zbMATHGoogle Scholar
  13. 13.
    Littlestone, N.: Learning Quickly When Irrelevant Attributes Abound: A New Linear-threshold Algorithm. Machine Learning 4(2), 285–318 (1988)Google Scholar
  14. 14.
    Littlestone, N.: From On-line to Batch Learning. In: Proc. 2nd COLT 1989, pp. 269–284 (1989)Google Scholar
  15. 15.
    Mannila, H., Räihä, K.: On the Complexity of Inferring Functional Dependencies. Discrete Applied Mathematics 40, 237–243 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Mossel, E., O’Donnell, R., Servedio, R.: Learning Juntas. In: Proc. STOC 2003, pp. 206–212 (2003)Google Scholar
  17. 17.
    Valiant, L.: Projection Learning. Machine Learning 37(2), 115–130 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jan Arpe
    • 1
  • Rüdiger Reischuk
    • 1
  1. 1.Institut für Theoretische InformatikUniversität zu LübeckLübeckGermany

Personalised recommendations