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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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