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Non-Adaptive Proper Learning Polynomials

Author Nader H. Bshouty

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  • Theory of computation
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  • Polynomial
  • Learning
  • Testing

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Abstract

We give the first polynomial-time non-adaptive proper learning algorithm of Boolean sparse multivariate polynomial under the uniform distribution. Our algorithm, for s-sparse polynomial over n variables, makes q = (s/ε)^{γ(s,ε)}log n queries where 2.66 ≤ γ(s,ε) ≤ 6.922 and runs in Õ(n)⋅ poly(s,1/ε) time. We also show that for any ε = 1/s^{O(1)} any non-adaptive learning algorithm must make at least (s/ε)^{Ω(1)}log n queries. Therefore, the query complexity of our algorithm is also polynomial in the optimal query complexity and optimal in n.

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Nader H. Bshouty. Non-Adaptive Proper Learning Polynomials. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.STACS.2023.16

Author Details

Nader H. Bshouty
  • Department of Computer Science, Technion, Haifa, Israel

References

  1. Dana Angluin. Queries and concept learning. Machine Learning, 2(4):319-342, 1987. Google Scholar
  2. Amos Beimel, Francesco Bergadano, Nader H. Bshouty, Eyal Kushilevitz, and Stefano Varricchio. Learning functions represented as multiplicity automata. J. ACM, 47(3):506-530, 2000. URL: https://doi.org/10.1145/337244.337257.
  3. Michael Ben-Or and Prasoon Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation (extended abstract). In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 301-309. ACM, 1988. URL: https://doi.org/10.1145/62212.62241.
  4. Francesco Bergadano, Nader H. Bshouty, and Stefano Varricchio. Learning multivariate polynomials from substitution and equivalence queries. Electron. Colloquium Comput. Complex., TR96-008, 1996. URL: https://eccc.weizmann.ac.il/eccc-reports/1996/TR96-008/index.html.
  5. Laurence Bisht, Nader H. Bshouty, and Hanna Mazzawi. On optimal learning algorithms for multiplicity automata. In Gábor Lugosi and Hans Ulrich Simon, editors, Learning Theory, 19th Annual Conference on Learning Theory, COLT 2006, Pittsburgh, PA, USA, June 22-25, 2006, Proceedings, volume 4005 of Lecture Notes in Computer Science, pages 184-198. Springer, 2006. URL: https://doi.org/10.1007/11776420_16.
  6. Avrim Blum and Mona Singh. Learning functions of k terms. In Mark A. Fulk and John Case, editors, Proceedings of the Third Annual Workshop on Computational Learning Theory, COLT 1990, University of Rochester, Rochester, NY, USA, August 6-8, 1990, pages 144-153. Morgan Kaufmann, 1990. URL: http://dl.acm.org/citation.cfm?id=92620.
  7. Nader H. Bshouty. On learning multivariate polynomials under the uniform distribution. Inf. Process. Lett., 61(6):303-309, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00021-5.
  8. Nader H. Bshouty. Testers and their applications. Electron. Colloquium Comput. Complex., page 11, 2012. URL: https://eccc.weizmann.ac.il/report/2012/011, URL: http://arxiv.org/abs/TR12-011.
  9. Nader H. Bshouty. Testers and their applications. In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 327-352. ACM, 2014. URL: https://doi.org/10.1145/2554797.2554828.
  10. Nader H. Bshouty. Almost optimal testers for concise representations. Electronic Colloquium on Computational Complexity (ECCC), 26:156, 2019. URL: https://eccc.weizmann.ac.il/report/2019/156, URL: http://arxiv.org/abs/TR19-156.
  11. Nader H. Bshouty. Almost optimal proper learning and testing polynomials. In Armando Castañeda and Francisco Rodríguez-Henríquez, editors, LATIN 2022: Theoretical Informatics - 15th Latin American Symposium, Guanajuato, Mexico, November 7-11, 2022, Proceedings, volume 13568 of Lecture Notes in Computer Science, pages 312-327. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-20624-5_19.
  12. Nader H. Bshouty and Yishay Mansour. Simple learning algorithms for decision trees and multivariate polynomials. SIAM J. Comput., 31(6):1909-1925, 2002. URL: https://doi.org/10.1137/S009753979732058X.
  13. Michael Clausen, Andreas W. M. Dress, Johannes Grabmeier, and Marek Karpinski. On zero-testing and interpolation of k-sparse multivariate polynomials over finite fields. Theor. Comput. Sci., 84(2):151-164, 1991. URL: https://doi.org/10.1016/0304-3975(91)90157-W.
  14. Arne Dür and Johannes Grabmeier. Applying coding theory to sparse interpolation. SIAM J. Comput., 22(4):695-704, 1993. URL: https://doi.org/10.1137/0222046.
  15. Paul Fischer and Hans Ulrich Simon. On learning ring-sum-expansions. SIAM J. Comput., 21(1):181-192, 1992. URL: https://doi.org/10.1137/0221014.
  16. Lisa Hellerstein and Rocco A. Servedio. On PAC learning algorithms for rich boolean function classes. Theor. Comput. Sci., 384(1):66-76, 2007. URL: https://doi.org/10.1016/j.tcs.2007.05.018.
  17. Ron M. Roth and Gyora M. Benedek. Interpolation and approximation of sparse multivariate polynomials over GF(2). SIAM J. Comput., 20(2):291-314, 1991. URL: https://doi.org/10.1137/0220019.
  18. Robert E. Schapire and Linda Sellie. Learning sparse multivariate polynomials over a field with queries and counterexamples. J. Comput. Syst. Sci., 52(2):201-213, 1996. URL: https://doi.org/10.1006/jcss.1996.0017.
  19. Leslie G. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, 1984. URL: https://doi.org/10.1145/1968.1972.
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