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Application of a Generalization of Russo's Formula to Learning from Multiple Random Oracles

Published online by Cambridge University Press:  09 July 2009

JAN ARPE  and
ELCHANAN MOSSEL
JAN ARPE
Affiliation:
Department of Statistics, UC Berkeley, CA 94720, USA and Bertelsmann Stiftung, Carl-Bertelsmann-Strasse 256, 33311 Gütersloh, Germany (e-mail: jan.arpe@bertelsmann.de)
ELCHANAN MOSSEL
Affiliation:
Departments of Statistics and Computer Science, UC Berkeley, CA 94720, USA and Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel (e-mail: mossel@stat.berkeley.edu)
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Abstract

We study the problem of learning k-juntas given access to examples drawn from a number of different product distributions. Thus we wish to learn a function f: {−1, 1}n → {−1, 1} that depends on k (unknown) coordinates. While the best-known algorithms for the general problem of learning a k-junta require running times of nk poly(n, 2k), we show that, given access to k different product distributions with biases separated by γ > 0, the functions may be learned in time poly(n, 2k, γ−k). More generally, given access to t ≤ k different product distributions, the functions may be learned in time nk/tpoly(n, 2k, γ−k). Our techniques involve novel results in Fourier analysis, relating Fourier expansions with respect to different biases, and a generalization of Russo's formula.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 2 , March 2010 , pp. 183 - 199
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Almuallim, H. and Dietterich, T. G. (1994) Learning Boolean concepts in the presence of many irrelevant features. Artificial Intelligence 69 279–305.Google Scholar
[2]Arpe, J. and Reischuk, R. (2007) Learning juntas in the presence of noise. Theoret. Comput. Sci. 384 2–21.Google Scholar
[3]Atıcı, A. and Servedio, R. A. (2007) Quantum algorithms for learning and testing juntas. Quantum Inf. Process. 6 323–348.Google Scholar
[4]Blum, A. (2003) Learning a function of r relevant variables. In Computational Learning Theory and Kernel Machines: COLT/Kernel 2003 (Schölkopf, B. and Warmuth, M K., eds), Vol. 2777 of Lecture Notes in Computer Science, Springer, pp. 731–733.Google Scholar
[5]Blum, A., Furst, M., Jackson, J. C., Kearns, M., Mansour, Y. and Rudich, S. (1994) Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In Proc. 26th Annual ACM Symposium on Theory of Computing: STOC '94 (Montreal), ACM Press, pp. 253–262.Google Scholar
[6]Blum, A. and Langley, P. (1997) Selection of relevant features and examples in machine learning. Artificial Intelligence 97 245–271.Google Scholar
[7]Blumer, A., Ehrenfeucht, A., Haussler, D. and Warmuth, M. K. (1989) Learnability and the Vapnik–Chervonenkis dimension. J. Assoc. Comput. Mach. 36 929–965.Google Scholar
[8]Bshouty, N. H. and Feldman, V. (2002) On using extended statistical queries to avoid membership queries. J. Mach. Learn. Res. 2 359–396.Google Scholar
[9]Bshouty, N. H. and Tamon, C. (1996) On the Fourier spectrum of monotone functions. J. Assoc. Comput. Mach. 43 747–770.Google Scholar
[10]Crammer, K., Kearns, M. and Wortman, J. (2005) Learning from data of variable quality. In Advances in Neural Information Processing Systems 18: NIPS 2005 (Vancouver), MIT Press.Google Scholar
[11]Crammer, K., Kearns, M. and Wortman, J. (2006) Learning from multiple sources. In Advances in Neural Information Processing Systems 19: NIPS '06 (Vancouver), MIT Press, pp. 321–328.Google Scholar
[12]Dooly, D. R., Zhang, Q., Goldman, S. A. and Amar, R. A. (2002) Multiple-instance learning of real-valued data. J. Mach. Learn. Res. 3 651–678.Google Scholar
[13]Feldman, J., O'Donnell, R. and Servedio, R. A. (2005) Learning mixtures of product distributions over discrete domains. In 46th Annual IEEE Symposium on Foundations of Computer Science: FOCS '05, IEEE Press, pp. 501–510.Google Scholar
[14]Furst, M. L., Jackson, J. C. and Smith, S. W. (1991) Improved learning of AC 0 functions. In Proc. 4th Annual Workshop on Computational Learning Theory: COLT 1991 (Valiant, L. G. and Warmuth, M. K., eds), Morgan Kaufmann, pp. 317–325.Google Scholar
[15]Grimmett, G. (1999) Percolation, 2nd edn, Grundlehren der mathematischen Wissenschaften, Springer,Google Scholar
[16]Hancock, T. R. and Mansour, Y. (1991) Learning monotone k-μ DNF formulas on product distributions. In Proc. 4th Annual Workshop on Computational Learning Theory: COLT 1991 (Valiant, L. G. and Warmuth, M. K., eds), Morgan Kaufmann, pp. 179–183.Google Scholar
[17]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.Google Scholar
[18]Kearns, M. (1998) Efficient noise-tolerant learning from statistical queries. J. Assoc. Comput. Mach. 45 983–1006.Google Scholar
[19]Kearns, M., Mansour, Y., Ron, D., Rubinfeld, R., Schapire, R. E. and Sellie, L. (1994) On the learnability of discrete distributions. In Proc. 26th Annual ACM Symposium on Theory of Computing: STOC '94 (Montreal), ACM Press, pp. 273–282.Google Scholar
[20]Kolountzakis, M. N., Markakis, E. and Mehta, A. (2005) Learning symmetric juntas in time no(k). In Workshop on Interface between Harmonic Analysis and Number Theory (Marseille 2005). Available as technical report at http://arxiv.org/abs/math.CO/0504246v1.Google Scholar
[21]Ling, C. X. and Yang, Q. (2006) Discovering classification from data of multiple sources. Data Min. Knowl. Discov. 12 181–201.Google Scholar
[22]Lipton, R. J., Markakis, E., Mehta, A. and Vishnoi, N. K. (2005) On the Fourier spectrum of symmetric Boolean functions with applications to learning symmetric juntas. In 20th Annual IEEE Conference on Computational Complexity: CCC '05, IEEE Computer Society Press, pp. 112–119.Google Scholar
[23]Mossel, E., O'Donnell, R. W. and Servedio, R. A. (2004) Learning functions of k relevant variables. J. Comput. System Sci. 69 421–434.Google Scholar
[24]Russo, L. (1981) On the critical percolation probabilities. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56 229–237.Google Scholar
[25]Servedio, R. A. (2004) On learning monotone DNF under product distributions. Inform. Comput. 193 57–74.Google Scholar
[26]Turán, G. (1993) Lower bounds for PAC learning with queries. In Proc. 6th Annual ACM Conference on Computational Learning Theory: COLT 1993 (Valiant, L. G. and Warmuth, M. K., eds), ACM Press, pp. 384–391.Google Scholar
[27]Valiant, L. G. (1984) A theory of the learnable. Commun. Assoc. Comput. Mach. 27 1134–1142.Google Scholar
[28]Vempala, S. and Wang, G. (2002) A spectral algorithm for learning mixtures of distributions. In 43rd Symposium on Foundations of Computer Science: FOCS 2002 (Vancouver), IEEE Press, p. 113.Google Scholar