Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Improved Bounds for Testing Juntas

  • Conference paper
Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (APPROX 2008, RANDOM 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5171))

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).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atıcı, A., Servedio, R.A.: Quantum algorithms for learning and testing juntas. Quantum Information Processing 6(5), 323–348 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and non-approximability – towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernstein, A.J.: Maximally connected arrays on the n-cube. SIAM J. Appl. Math. 15(6), 1485–1489 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blum, A.: Relevant examples and relevant features: thoughts from computational learning theory. In: AAAI Fall Symposium on ‘Relevance’ (1994)

    Google Scholar 

  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. Blum, A., Langley, P.: Selection of relevant features and examples in machine learning. Artificial Intelligence 97(2), 245–271 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bollobás, B.: Combinatorics, Cambridge (1986)

    Google Scholar 

  8. Chockler, H., Gutfreund, D.: A lower bound for testing juntas. Information Processing Letters 90(6), 301–305 (2004)

    Article  MathSciNet  Google Scholar 

  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. Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Comput. Syst. Sci. 68(4), 753–787 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Harper, L.H.: Optimal assignments of numbers to vertices. SIAM J. Appl. Math. 12(1), 131–135 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hart, S.: A note on the edges of the n-cube. Disc. Math. 14, 157–163 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. Mossel, E., O’Donnell, R., Servedio, R.A.: Learning functions of k relevant variables. J. Comput. Syst. Sci. 69(3), 421–434 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Discret. Math. 16(1), 20–46 (2003)

    Article  MathSciNet  Google Scholar 

  19. Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Carnegie Mellon University,  

    Eric Blais

Authors
  1. Eric Blais
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ashish Goel Klaus Jansen José D. P. Rolim Ronitt Rubinfeld

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Blais, E. (2008). Improved Bounds for Testing Juntas. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-85363-3_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85362-6

  • Online ISBN: 978-3-540-85363-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation