Noise-Tolerant Distribution-Free Learning of General Geometric Concepts

Journal of the ACM

Volumn 45, Issue 5, 1998, Pages 863-890

  Bshouty, Nader H ^a,f   Goldman, Sally A ^b   Mathias, H David ^c,e   Suri, Subhash ^b   Tamaki, Hisao ^d  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

^b WASHINGTON UNIVERSITY   (United States)

^c OHIO STATE UNIVERSITY   (United States)

^d IBM RESEARCH TOKYO   (Japan)

^e UNIVERSITY OF WASHINGTON   (United States)

^f UNIVERSITY OF CALGARY   (Canada)

Author keywords

Algorithms; Computational learning; F.2.2 Analysis of Algorithms and Problem Complexity : Nonnumerical Algorithms and Problems; Geometric concepts; Theory

Indexed keywords

ALGORITHMS; BOOLEAN FUNCTIONS; COMPUTATIONAL GEOMETRY; POLYNOMIALS; RANDOM PROCESSES; STATISTICAL METHODS;

COMPUTATIONAL LEARNING;

LEARNING SYSTEMS;

EID: 0032157440     PISSN: 00045411     EISSN: None     Source Type: Journal    
DOI: 10.1145/290179.290184     Document Type: Article

References (37)

1. ANGLUIN, D. 1988. Queries and concept learning. Mach. Learn. 2, 4, 319-342.
2. ANGLUIN, D., AND LAIRD, P. 1988. Learning from noisy examples. Mach. Learn. 2, 4, 343-370.
3. ASLAM, J. A., AND DECATUR, S. E. 1993. General bounds on statistical query learning and PAC learning with noise via hypothesis boosting. In Proceedings of the 34th Annual Symposium on Foundations of Computer Science (Nov.). pp. 282-291.
4. ASLAM, J. A., AND DECATUR, S. E. 1995. Specification and simulation of statistical query algorithms for efficiency and noise tolerance. In Proceedings of the 8th Annual ACM Conference on Computational Learning Theory (Santa Cruz, Calif., July 5-8). ACM, New York, pp. 437-446.
5. AUER, P., KWEK, S., MAASS, W., AND WARMUTH, M. K. 1996. Learning of depth two neural networks with constant fan-in at the hidden nodes. In Proceedings of the 9th Annual ACM Conference on Computational Learning Theory (Desenzano del Garda, Italy, June 28-July 1). ACM, New York, pp. 333-343.
6. BAUM, E. B. 1990a. On learning a union of half spaces. J. Complexity 6, 1 (March), 67-101.
7. BAUM, E. B. 1990b. The perceptron algorithm is fast for nonmalicious distributions. Neural Comput. 2, 248-260.
8. BAUM, E. B. 1991. Neural net algorithms that learn in polynomial time from examples and queries. IEEE Trans. Neural Netw. 2, 1 (Jan.), 5-19.
9. BEN-DAVID, S., BSHOUTY, N., AND KUSHILEVITZ, E. 1997. A composition theorem for learning algorithms with applications to geometric concept classes. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing (El Paso, Tex. May 4-6). ACM, New York, pp. 324-333.
10. BERGER, B., ROMPEL, J., AND SHOR, P. W. 1994. Efficient NC algorithms for set cover with applications to learning and geometry. J. Comput. Syst. Sci. 49, 2, 454-477.
11. BLUM, A., AND RIVEST, R. L. 1989. Training a 3-node neural net is NP-Complete. In Advances in Neural Information Processing Systems I. Morgan-Kaufmann, San Mateo, Calif., pp. 494-501.
12. BLUMER, A., EHRENFEUCHT, A., HAUSSLER, D., AND WARMUTH, M. K. 1987. Occam's razor. Inf. Proc. Lett. 24, 377-380.
13. BLUMER, A., EHRENFEUCHT, A., HAUSSLER, D., AND WARMUTH, M. K. 1989. Learnability and the Vapnik-Chervonenkis dimension. J. ACM 36, 4 (Oct.), 929-965.
14. BRÖNNIMANN, H., AND GOODRICH, M. T. 1994. Almost optimal set covers in finite VC-dimension. In Proceedings of the 10th Annual Symposium on Computational Geometry (May). pp. 293-302.
15. BSHOUTY, N. 1998. A new composition theorem for learning algorithms. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (Dallas, Tex., May 23-26). ACM, New York, pp. 583-589.
16. BSHOUTY, N., CHEN, Z., AND HOMER, S. 1994. On learning discretized geometric concepts. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science (Oct.). IEEE Computer Science Press, Los Alamitos, Calif., pp. 54-63.
17. BSHOUTY, N. H., GOLDBERG, P. W., GOLDMAN, S. A., AND MATHIAS, H. D. 1999. Exact learning of discretized geometric concepts. SIAM J. Comput. 28, 2, 674-699.
18. BSHOUTY, N. H., GOLDMAN, S. A., AND MATHIAS, H. D. 1998. Noise-tolerant parallel learning of geometric concepts. Inf. Comput., to appear.
19. CHEN, Z., AND HOMER, S. 1994. The bounded injury priority method and the learnability of unions of rectangles. Unpublished manuscript.
20. CHVATAL, V. 1979. A greedy heuristic for the set covering problem. Math. Op. Res. 4, 3, 233-235.
21. DECATUR, S. E. 1993. Statistical queries and faulty PAC oracles. In Proceedings of the 6th Annual ACM Conference on Computational Learning Theory (Santa Cruz, Calif., July 26-28). ACM, New York, pp. 262-268.
22. EDELSBRUNNER, H. 1987. Algorithms in Combinatorial Geometry. Springer-Verlag, New York.
23. EHRENFEUCHT, A., AND HAUSSLER, D. 1989. Learning decision trees from random examples. Inf. Comput. 82, 3 (Sept.), 231-246.
24. FRAZIER, M., GOLDMAN, S., MISHRA, N., AND PITT, L. 1996. Learning from a consistently ignorant teacher. J. Comput. Syst. Sci. 52, 3 (June), 472-492.
25. GOLDBERG, P. W., GOLDMAN, S. A., AND MATHIAS, H. D. 1993. Learning unions of rectangles with membership and equivalence queries. Tech. Rep. WUCS-93-46 (Nov.), Washington Univ., St. Louis, MO.
26. HAUSSLER, D. 1989. Generalizing the PAC model: sample size bounds from metric dimension-based uniform convergence results. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science (Oct.). IEEE Computer Society Press, Los Alamitos, Calif., pp. 40-45.
27. HAUSSLER, D., AND WELZL, E. 1987. Epsilon-nets and simplex range queries. Disc. Comput. Geom. 2, 127-151.
28. JOHNSON, D. S. 1974. Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 256-278.
29. KEARNS, M. 1993. Efficient noise-tolerant learning from statistical queries. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 392-401.
30. KWEK, S., AND PITT, L. 1996. PAC learning intersections of halfspaces with membership queries. In Proceedings of the 9th Annual ACM Conference on Computational Learning Theory (Desenzuno del Garda, Italy, June 28-July 1). ACM, New York, pp. 244-254.
31. LONG, P. M., AND WARMUTH, M. K. 1994. Composite geometric concepts and polynomial predictability. Inf. Comput. 113, 2, 203-252.
32. LOVÁSZ, L. 1975. On the ratio of optimal integral and fractional covers. Disc. Math. 13, 383-390.
33. SLOAN, R. H. 1995. Four types of noise in data for PAC learning. Inf. Proc. Lett. 54, 157-162.
34. VALIANT, L. G. 1984. A theory of the learnable. Commun. ACM 27, 11 (Nov.), 1134-1142.
35. VALIANT, L. 1985. Learning disjunctions of conjunctions. In Proceedings of 9th International Joint Conference on Artificial Intelligence.
36. VAPNIK, V. N., AND CHERVONENKIS, A. Y. 1971. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and Its Applications XVI, 2, 264-280.
37. WILLARD, D. E. 1982. Polygon retrieval. SIAM J. Comput. 11, 1 (Feb.), 149-165.