Abstract
We assume wlog. that every learning algorithm with membership and equivalence queries proceeds in rounds. In each round it puts in parallel a polynomial number of queries and after receiving the answers, it performs internal computations before starting the next round. The query depth is defined by the number of rounds. In this paper we show that, assuming the existence of cryptographic one-way functions, for any fixed polynomial d (n) there exists a concept class that is efficiently and exactly learnable with membership queries in query depth d(n) +1, but cannot be weakly predicted with membership and equivalence queries in depth d(n). Hence, concerning the query depth, efficient learning algorithms for this concept class cannot be parallelized at all. We also discuss some applications to random self-reductions and coherent sets.
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
D. Angluin: Learning Regular Sets from Queries and Counterexamples, Informa-tion and Computation, vol. 75, pp. 87–106, 1987.
D. Angluin: Queries and Concept Learning, Machine Learning, vol. 2, pp. 319–342, 1988.
D. Angluin, M. Kharitonov: When Won’t Membership Queries Help?, 23rd ACM Symposium on Theory of Computing, pp. 444–454, 1991.
J. Balcázar, J. Díaz, R. Gavaldà, O. Watanabe: An Optimal Parallel Algorithm for Learning DFA, 7th ACM Conference on Computational Learning Theory 1994.
R. Beigel, J. Feigenbaum: Improved Bounds on Coherence and Checkability, Yale University Technical Report, YALEU/DCS/TR-819, 1990.
R. Beigel, J. Feigenbaum: On Being Incoherent Without Being very Hard, Com-putational Complexity, vol. 2, pp. 1–17, 1992.
M. Bellare, S. Goldwasser: The Complexity of Decision Versus Search, SIAM Journal on Computing, vol. 23, no. 1, 1994.
M. Blum, S. Micali: How to Generate Cryptographically Strong Sequences of Pseudorandom Bits, SIAM Journal on Computation, vol. 13, pp. 850–864, 1984.
N. Bshouty: Exact Learning of Formulas in Parallel, Machine Learning, vol. 26, pp. 25–42, 1997.
N. Bshouty, R. Cleve: On the Exact Learning of Formulas in Parallel, 33rd IEEE Symposium on the Foundations of Computer Science, pp. 513–522, 1992.
N. Bshouty, S. Goldman, H. Mathias: Noise-Tolerant Parallel Learning of Geometric Concepts, 8th ACM Conference on Computational Learning Theory, 1995.
B. Berger, J. Rompel, P. Shor: Efficient NC Algorithms for Set Cover with Application to Learning and Geometry, 30th IEEE Symposium on the Foundations of Computer Science, pp. 54–59, 1989.
R. Canetti, D. Micciancio, O. Reingold: Perfectly One-Way Probabilistic Hash Functions, 30th ACM Symposium on Theory of Computing, 1998.
J. Feigenbaum, L. Fortnow: On the Random-Self-Reducibility of Complete Sets, 6th Annual IEEE Structure in Complexity Theory Conference, 1991.
J. Feigenbaum, L. Fortnow, C. Lund, D. Spielman: The Power of Adaptiveness and Additional Queries in Random-Self-Reductions, Computational Complexity, vol. 4, pp. 158–174, 1994.
S. Goldwasser, O. Goldreich, S. Micali: How to Construct Random Functions, Journal of ACM, vol. 33, pp. 792–807, 1986.
J. Håstad, R. Impagliazzo, L. Levin, M. Luby: Construction of Pseudorandom Generator from any One-Way Function, to appear in SIAM Journal on Computing, preliminary versions in STOC’89 and STOC’90, 1989/1990.
E. Hemaspaandra, A. Naik, M. Ogihara, A. Selman: P-Selective Sets, and Reducing Search to Decision vs.Self-Reducibility, Journal of Computer and System Sciences, vol. 53, pp. 194–209, 1996.
R. Impagliazzo, M. Luby: One-Way Functions are Essential for Complexity Based Cryptography, 30th IEEE Symposium on Foundations of Computer Science, pp. 230–235, 1989.
M. Kearns, L. Valiant: Cryptographic Limitations on Learning Boolean Formulae and Finite Automata, Journal of ACM, vol. 41, pp. 67–95, 1994.
M. Kharitonov: Cryptographic Hardness of Distribution-Specific Learning, 25th ACM Symposium on the Theory of Computing, pp. 372–381, 1993.
R. Ostrovsky, A. Wigderson: One-Way Functions are Essential for Non-Trivial Zero-Knowledge, Second IEEE Israel Symposium on Theory and Computing Sys-tems, 1993.
L. Pitt, L. Valiant: Computational Limitations on Learning from Examples, Journal of ACM, vol. 35, pp. 965–984, 1988.
L. Pitt, M. Warmuth: Prediction-Preserving Reducibility, Journal of Computer and System Science, vol. 41, pp. 430–467, 1990.
R. Rivest, Y.L. Lin: Being Taught can be Faster than Asking Questions, 8th ACM Conference on Computational Learning Theory, 1995.
C.P. Schnorr: Optimal Algorithms for Self-Reducible Problems, 3rd International Colloqium on Automata, Languages, and Programming, pp. 322–347, Edingburgh University Press, 1976.
L. Valiant: A Theory of the Learnable, Communications of ACM, vol. 27, pp. 1134–1142, 1984.
J. Vitter, J. Lin: Learning in Parallel, Information and Computation, vol. 92, pp. 179–202, 1992.
A. Yao: Coherent Functions and Program Checkers, 22nd ACM Symposium on Theory of Computing, pp. 84–94, 1990.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fischlin, M. (1998). Cryptographic Limitations on Parallelizing Membership and Equivalence Queries with Applications to Random Self-Reductions. In: Richter, M.M., Smith, C.H., Wiehagen, R., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 1998. Lecture Notes in Computer Science(), vol 1501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-49730-7_6
Download citation
DOI: https://doi.org/10.1007/3-540-49730-7_6
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65013-3
Online ISBN: 978-3-540-49730-1
eBook Packages: Springer Book Archive