Multiple Random Oracles Are Better Than One

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learning juntas
PAC learning
biased product distributions
Fourier analysis of Boolean functions
Russo’s formula
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Statistics and Probability
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Arpe, Jan
Mossel, Elchanan
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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.

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2010-03-01
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Combinatorics, Probability and Computing
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At the time of publication, author Elchanan Mossel was affiliated with the University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania. The postprint version of this article, Multiple Random Oracles Are Better Than One, is published in its final form under the title Application of a Generalization of Russo's Formula to Learning from Multiple Random Oracles.
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