Learning intersections of halfspaces with a margin

Journal of Computer and System Sciences

Volumn 74, Issue 1, 2008, Pages 35-48

  Klivans, Adam R ^a   Servedio, Rocco A ^b  

^a University of Texas at Austin   (United States)

^b Columbia University ^*  (United States)

Author keywords

Computational learning theory; Intersections of halfspaces; Kernel Perceptron; Margin; Polynomial threshold function; Random projection

Indexed keywords

LEARNING ALGORITHMS; NUMERICAL METHODS; POLYNOMIALS;

COMPUTATIONAL LEARNING THEORY; INTERSECTIONS OF HALFSPACES; KERNEL PERCEPTRON; POLYNOMIAL THRESHOLD FUNCTIONS; RANDOM PROJECTION;

LEARNING SYSTEMS;

EID: 35448932401     PISSN: 00220000     EISSN: 10902724     Source Type: Journal    
DOI: 10.1016/j.jcss.2007.04.012     Document Type: Article

References (29)

1. Achlioptas D. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. Comput. System Sci. 66 4 (2003) 671-687
2. Anthony M. Classification by polynomial surfaces. Discrete Appl. Math. 61 (1995) 91-103
3. R. Arriaga, S. Vempala, An algorithmic theory of learning: Robust concepts and random projection, in: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, FOCS, 1999, pp. 616-623
4. Baum E. A polynomial time algorithm that learns two hidden unit nets. Neural Comput. 2 (1991) 510-522
5. Beigel R. When do extra majority gates help? polylog (n) majority gates are equivalent to one. Comput. Complexity 4 (1994) 314-324
6. Beigel R., Reingold N., and Spielman D. PP is closed under intersection. J. Comput. System Sci. 50 2 (1995) 191-202
7. Blum A., and Kannan R. Learning an intersection of a constant number of halfspaces under a uniform distribution. J. Comput. System Sci. 54 2 (1997) 371-380
8. Blumer A., Ehrenfeucht A., Haussler D., and Warmuth M. Learnability and the Vapnik-Chervonenkis dimension. J. ACM 36 4 (1989) 929-965
9. Cheney E. Introduction to Approximation Theory (1966), McGraw-Hill, New York, NY
10. Cristianini N., and Shawe-Taylor J. An Introduction to Support Vector Machines (and Other Kernel-Based Learning Methods) (2000), Cambridge University Press
11. S. Dasgupta, Learning mixtures of Gaussians, in: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, 1999, pp. 634-644
12. Y. Freund, R. Schapire, Large margin classification using the Perceptron algorithm, in: Proceedings of the Eleventh Annual Conference on Computational Learning Theory, 1998, pp. 209-217
13. Herbrich R. Learning Kernel Classifiers (2002), MIT Press
14. Johnson W., and Lindenstrauss J. Extensions of Lipshitz mapping into Hilbert space. Contemp. Math. vol. 26 (1984) 189-206
15. Khardon R., Roth D., and Servedio R. Efficiency versus convergence of Boolean kernels for on-line learning algorithms. In: Dietterich T.G., Becker S., and Ghahramani Z. (Eds). Advances in Neural Information Processing Systems vol. 14 (2002), MIT Press, Cambridge, MA
16. Klivans A., O'Donnell R., and Servedio R. Learning intersections and thresholds of halfspaces. Proceedings of the Forty-Third Annual Symposium on Foundations of Computer Science (2002) 177-186
17. 1 / 3), in: Proceedings of the Thirty-Third Annual Symposium on Theory of Computing, 2001, pp. 258-265
18. Newman D.J. Rational approximation to | x |. Michigan Math. J. 11 (1964) 11-14
19. A. Novikoff, On convergence proofs on perceptrons, in: Proceedings of the Symposium on Mathematical Theory of Automata, vol. XII, 1962, pp. 615-622
20. R. O'Donnell, R. Servedio, Extremal properties of polynomial threshold functions, in: Proceedings of the Eighteenth Annual Conference on Computational Complexity, 2003, pp. 3-12
21. R. O'Donnell, R. Servedio, New degree bounds for polynomial threshold functions, in: Proceedings of the 35th ACM Symposium on Theory of Computing, 2003, pp. 325-334
22. Rosenblatt F. Principles of Neurodynamics (1962), Springer-Verlag, New York
23. Saks M. Slicing the hypercube. London Math. Soc. Lecture Note Ser. vol. 187 (1993) 211-257
24. A. Shpilka, Lower bounds for small depth arithmetic and Boolean circuits, PhD thesis, Hebrew University, 2001
25. Vempala S. A random sampling based algorithm for learning the intersection of halfspaces. available at. http://www-math.mit.edu/~vempala/papers/robust.ps A preliminary version appeared in: Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, pp. 508-513
26. Valiant L. A theory of the learnable. Commun. ACM 27 11 (1984) 1134-1142
27. S. Vempala, personal communication, 2004
28. Vempala S. The Random Projection Method (2004), American Mathematical Society DIMACS
29. Zhou J. Introduction to symmetric polynomials and symmetric functions. Lecture Notes for Course at Tsinghua University, available at (2003). http://cms.zju.edu.cn/course/cn/SymmetricF.pdf