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Abstract

We consider the problem of testing functions for the property of being a k-junta (i.e., of depending on at most k variables). Fischer, Kindler, Ron, Safra, and Samorodnitsky (J. Comput. Sys. Sci., 2004) showed that \(\tilde{O}(k^2)/\epsilon\) queries are sufficient to test k-juntas, and conjectured that this bound is optimal for non-adaptive testing algorithms.

Our main result is a non-adaptive algorithm for testing k-juntas with \(\tilde{O}(k^{3/2})/\epsilon\) queries. This algorithm disproves the conjecture of Fischer et al.

We also show that the query complexity of non-adaptive algorithms for testing juntas has a lower bound of \(\min \big(\tilde{\Omega}(k/\epsilon), 2^k/k\big)\), essentially improving on the previous best lower bound of Ω(k).

Keywords

Boolean Function Input Function Block Test Query Complexity Statistical Distance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Atıcı, A., Servedio, R.A.: Quantum algorithms for learning and testing juntas. Quantum Information Processing 6(5), 323–348 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and non-approximability – towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bernstein, A.J.: Maximally connected arrays on the n-cube. SIAM J. Appl. Math. 15(6), 1485–1489 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Blum, A.: Relevant examples and relevant features: thoughts from computational learning theory. In: AAAI Fall Symposium on ‘Relevance’ (1994)Google Scholar
  5. 5.
    Blum, A.: Learning a function of r relevant variables. In: Proc. 16th Conference on Computational Learning Theory, pp. 731–733 (2003)Google Scholar
  6. 6.
    Blum, A., Langley, P.: Selection of relevant features and examples in machine learning. Artificial Intelligence 97(2), 245–271 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bollobás, B.: Combinatorics, Cambridge (1986)Google Scholar
  8. 8.
    Chockler, H., Gutfreund, D.: A lower bound for testing juntas. Information Processing Letters 90(6), 301–305 (2004)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Diakonikolas, I., Lee, H.K., Matulef, K., Onak, K., Rubinfeld, R., Servedio, R.A., Wan, A.: Testing for concise representations. In: Proc. 48th Symposium on Foundations of Computer Science, pp. 549–558 (2007)Google Scholar
  10. 10.
    Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Comput. Syst. Sci. 68(4), 753–787 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gonen, M., Ron, D.: On the benefits of adaptivity in property testing of dense graphs. In: Proc. 11th Workshop RANDOM, pp. 525–539 (2007)Google Scholar
  12. 12.
    Guijarro, D., Tarui, J., Tsukiji, T.: Finding relevant variables in PAC model with membership queries. In: Proc. 10th Conference on Algorithmic Learning Theory, pp. 313–322 (1999)Google Scholar
  13. 13.
    Harper, L.H.: Optimal assignments of numbers to vertices. SIAM J. Appl. Math. 12(1), 131–135 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hart, S.: A note on the edges of the n-cube. Disc. Math. 14, 157–163 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kahn, J., Kalai, G., Linial, N.: The influence of variables on boolean functions. In: Proc. 29th Sym. on Foundations of Computer Science, pp. 68–80 (1988)Google Scholar
  16. 16.
    Lipton, R.J., Markakis, E., Mehta, A., Vishnoi, N.K.: On the Fourier spectrum of symmetric boolean functions with applications to learning symmetric juntas. In: Proc. 20th Conference on Computational Complexity, pp. 112–119 (2005)Google Scholar
  17. 17.
    Mossel, E., O’Donnell, R., Servedio, R.A.: Learning functions of k relevant variables. J. Comput. Syst. Sci. 69(3), 421–434 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Discret. Math. 16(1), 20–46 (2003)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Yao, A.C.: Probabilistic computations: towards a unified measure of complexity. In: Proc. 18th Sym. on Foundations of Comput. Sci., pp. 222–227 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Eric Blais
    • 1
  1. 1.Carnegie Mellon University 

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