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
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Atıcı, A., Servedio, R.A.: Quantum algorithms for learning and testing juntas. Quantum Information Processing 6(5), 323–348 (2007)
Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs and non-approximability – towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)
Bernstein, A.J.: Maximally connected arrays on the n-cube. SIAM J. Appl. Math. 15(6), 1485–1489 (1967)
Blum, A.: Relevant examples and relevant features: thoughts from computational learning theory. In: AAAI Fall Symposium on ‘Relevance’ (1994)
Blum, A.: Learning a function of r relevant variables. In: Proc. 16th Conference on Computational Learning Theory, pp. 731–733 (2003)
Blum, A., Langley, P.: Selection of relevant features and examples in machine learning. Artificial Intelligence 97(2), 245–271 (1997)
Bollobás, B.: Combinatorics, Cambridge (1986)
Chockler, H., Gutfreund, D.: A lower bound for testing juntas. Information Processing Letters 90(6), 301–305 (2004)
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)
Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Comput. Syst. Sci. 68(4), 753–787 (2004)
Gonen, M., Ron, D.: On the benefits of adaptivity in property testing of dense graphs. In: Proc. 11th Workshop RANDOM, pp. 525–539 (2007)
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)
Harper, L.H.: Optimal assignments of numbers to vertices. SIAM J. Appl. Math. 12(1), 131–135 (1964)
Hart, S.: A note on the edges of the n-cube. Disc. Math. 14, 157–163 (1976)
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)
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)
Mossel, E., O’Donnell, R., Servedio, R.A.: Learning functions of k relevant variables. J. Comput. Syst. Sci. 69(3), 421–434 (2004)
Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Discret. Math. 16(1), 20–46 (2003)
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comput. 25(2), 252–271 (1996)
Yao, A.C.: Probabilistic computations: towards a unified measure of complexity. In: Proc. 18th Sym. on Foundations of Comput. Sci., pp. 222–227 (1977)
Author information
Authors and Affiliations
Editor information
Rights 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)