Abstract
We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function f: {0,1}n →{0,1} is an s-sparse GF(2) polynomial versus ε-far from every such polynomial. Our algorithm makes poly(s,1/ε) black-box queries to f and runs in time n ·poly(s,1/ε). The only previous algorithm for this testing problem [DLM + 07] used poly(s,1/ε) queries, but had running time exponential in s and super-polynomial in 1/ε.
Our approach significantly extends the “testing by implicit learning” methodology of [DLM + 07]. The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie [SS96]. A crucial element of this work, which enables us to simulate the membership queries required by [SS96], is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of “low-influence” sets of variables.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing low-degree polynomials over GF(2). In: Proc. RANDOM, pp. 188–199 (2003)
Angluin, D.: Queries and concept learning. Machine Learning 2, 319–342 (1988)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comp. Sys. Sci 47, 549–595 (1993); Earlier version in STOC 1990
Bshouty, N., Mansour, Y.: Simple Learning Algorithms for Decision Trees and Multivariate Polynomials. SIAM J. Comput. 31(6), 1909–1925 (2002)
Blum, A., Singh, M.: Learning functions of k terms. In: Proceedings of the 3rd Annual Workshop on Computational Learning Theory (COLT), pp. 144–153 (1990)
Bshouty, N.: On learning multivariate polynomials under the uniform distribution. Information Processing Letters 61(3), 303–309 (1997)
Bshouty, N.: Simple learning algorithms using divide and conquer. Computational Complexity 6, 174–194 (1997)
Diakonikolas, I., Lee, H., Matulef, K., Onak, K., Rubinfeld, R., Servedio, R., Wan, A.: Testing for concise representations. In: Proc. 48th Ann. Symposium on Computer Science (FOCS), pp. 549–558 (2007)
Ehrenfeucht, A., Karpinski, M.: The computational complexity of (xor,and)-counting problems. Technical report (preprint 1989)
Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. Journal of Computer & System Sciences 68, 753–787 (2004)
Fischer, P., Simon, H.U.: On learning ring-sum expansions. SIAM Journal on Computing 21(1), 181–192 (1992)
Goldreich, O., Goldwaser, S., Ron, D.: Property testing and its connection to learning and approximation. Journal of the ACM 45, 653–750 (1998)
Grigoriev, D., Karpinski, M., Singer, M.: Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal on Computing 19(6), 1059–1063 (1990)
Karpinski, M.: Boolean circuit complexity of algebraic interpolation problems (TR-89-027) (1989)
Kahn, J., Kalai, G., Linial, N.: The influence of variables on boolean functions. In: Proc. 29th FOCS, pp. 68–80 (1988)
Karpinski, M., Luby, M.: Approximating the Number of Zeros of a GF[2] Polynomial. Journal of Algorithms 14, 280–287 (1993)
Luby, M., Velickovic, B., Wigderson, A.: Deterministic approximate counting of depth-2 circuits. In: Proceedings of the 2nd ISTCS, pp. 18–24 (1993)
Mansour, Y.: Randomized interpolation and approximation of sparse polynomials. SIAM Journal on Computing 24(2), 357–368 (1995)
Matulef, K., O’Donnell, R., Rubinfeld, R., Servedio, R.: Testing Halfspaces. Technical Report 128, Electronic Colloquium in Computational Complexity (2007)
Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Disc. Math. 16, 20–46 (2002)
Roth, R., Benedek, G.: Interpolation and approximation of sparse multivariate polynomials over GF(2). SIAM J. Comput. 20(2), 291–314 (1991)
Ron, D.: Property testing: A learning theory perspective. In: Bshouty, N.H., Gentile, C. (eds.) COLT. LNCS (LNAI), vol. 4539. Springer, Heidelberg (2007), http://www.eng.tau.ac.il/~danar/Public-ppt/colt07.ppt
Schapire, R., Sellie, L.: Learning sparse multivariate polynomials over a field with queries and counterexamples. J. Comput. & Syst. Sci. 52(2), 201–213 (1996)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Diakonikolas, I., Lee, H.K., Matulef, K., Servedio, R.A., Wan, A. (2008). Efficiently Testing Sparse GF(2) Polynomials. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-70575-8_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70574-1
Online ISBN: 978-3-540-70575-8
eBook Packages: Computer ScienceComputer Science (R0)