ScholarlyCommons

Title

Multiple Random Oracles Are Better Than One

Author(s)

Jan Arpe
Elchanan Mossel, University of Pennsylvania

Document Type

Journal Article

Date of this Version

3-2010

Publication Source

Combinatorics, Probability and Computing

Volume

19

Issue

2

Start Page

183

Last Page

199

DOI

10.1017/S0963548309990277

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 n^k poly(n, 2^k), 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, 2^k, γ^−k). More generally, given access to t ≤ k different product distributions, the functions may be learned in time n^k/tpoly(n, 2^k, 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.

Comments

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.

Keywords

learning juntas, PAC learning, biased product distributions, Fourier analysis of Boolean functions, Russo’s formula

Recommended Citation

Arpe, J., & Mossel, E. (2010). Multiple Random Oracles Are Better Than One. Combinatorics, Probability and Computing, 19 (2), 183-199. http://dx.doi.org/10.1017/S0963548309990277