Abstract
We study polynomial time learning algorithms for Multiplicity Automata (MA) and Multiplicity Automata Function (MAF) that minimize the access to one or more of the following resources: Equivalence queries, Membership queries or Arithmetic operations in the field \({\cal F}\). This is in particular interesting when access to one or more of the above resources is significantly more expensive than the others.
We apply new algebraic approach based on Matrix Theory to simplify the algorithms and the proofs of their correctness. We improve the arithmetic complexity of the problem and argue that it is almost optimal. Then we prove tight bound for the minimal number of equivalence queries and almost (up to log factor) tight bound for the number of membership queries.
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.
References
Angluin, D.: Learning regular sets from queries and counterexamples. Information and Computation 75(2), 87–106 (1987)
Angluin, D.: Queries and concept learning. Machine Learning 2, 319–342 (1987)
Bläser, M.: Lower bounds for the multiplicative complexity of matrix multiplication. Comput. Complexity 8(3), 203–226 (1999)
Beimel, A., Bergadano, F., Bshouty, N.H., Kushilevitz, E., Varricchio, S.: Learning functions represented as multiplicity automata. J. ACM 47(3), 506–530 (2000)
Bergadano, F., Bshouty, N.H., Tamon, C., Varricchio, S.: On Learning Programs and Small Depth Circuits. In: Ben-David, S. (ed.) EuroCOLT 1997. LNCS, vol. 1208, pp. 150–161. Springer, Heidelberg (1997)
Bergadano, F., Bshouty, N.H., Varricchio, S.: Learning Multivariate Polynomials from Substitution and Equivalence Queries (1996), http://www.eccc.uni-trier.de/eccc
Bergadano, F., Catalano, D., Varricchio, S.: Learning Sat-k-DNF Formulas from Membership Queries. In: STOC 1996, pp. 126–130 (1996)
Balcázar, J.L., Díaz, J., Gavaldá, R., Watanabe, O.: A note on the query complexity of learning DFA. In: Doshita, S., Nishida, T., Furukawa, K., Jantke, K.P. (eds.) ALT 1992. LNCS, vol. 743. Springer, Heidelberg (1993)
Bshouty, N.H., Goldman, S.A., Hancock, T.R., Matar, S.: Asking Questions to Minimize Errors. J. Comput. Syst. Sci. 52(2), 268–286 (1996)
Beimel, A., Kushilevitz, E.: Learning Boxes in High Dimension. Algorithmica 22(1/2), 76–90 (1998)
Bshouty, N.H., Tamon, C., Wilson, D.K.: Learning matrix functions over rings. Algorithmica 22(1/2), 91–111 (1998)
Bergadano, F., Varricchio, S.: Learning Behaviors of Automata from Multiplicity and Equivalence Queries. SIAM J. Comput 25(6), 1268–1280 (1996)
Bergadano, F., Varricchio, S.: Learning behaviors of automata from shortest counterexamples. In: Vitányi, P.M.B. (ed.) EuroCOLT 1995. LNCS, vol. 904, pp. 380–391. Springer, Heidelberg (1995)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation 9, 251–280 (1990)
Elkies, N.: On finite sequences satisfying linear recursions. Available at: http://xxx.lanl.gov/abs/math.CO/0105007
Kearns, M.J., Valiant, L.G.: Cryptographic Limitations on Learning Boolean Formulae and Finite Automata. J. ACM 41(1), 67–95 (1994)
Littlestone, N.: Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning 2, 285–318 (1987)
Littlestone, N., Warmuth, M.K.: The weighted majority algorithm. Information and Computation 108(2), 212–261 (1994)
Maass, W., Turán, G.: Lower bound methods and seperation results for on-line learning models. Machine Learning 9, 107–145 (1992)
Ohnishi, H., Seki, H., Kasami, T.: A polynomial time learning algorithm for recognizable series. IEICE Transactions on Information and Systems E77-D(5), 1077–1085 (1994)
Shpilka, A.: Lower Bounds for Matrix Product. SIAM J. Comput. 32(5), 1185–1200 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bisht, L., Bshouty, N.H., Mazzawi, H. (2006). On Optimal Learning Algorithms for Multiplicity Automata. In: Lugosi, G., Simon, H.U. (eds) Learning Theory. COLT 2006. Lecture Notes in Computer Science(), vol 4005. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11776420_16
Download citation
DOI: https://doi.org/10.1007/11776420_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35294-5
Online ISBN: 978-3-540-35296-9
eBook Packages: Computer ScienceComputer Science (R0)