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Multiple Random Oracles Are Better Than One

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 time of n^k * poly(n,2^k), we show that given access to k different product distributions with biases separated by \gamma>0, the functions may be learned in time poly(n,2^k,\gamma^{-k}). More generally, given access to t <= k different product distributions, the functions may be learned in time n^{k/t} * poly(n,2^k,\gamma^{-k}). Our techniques involve novel results in Fourier analysis relating Fourier expansions with respect to different biases and a generalization of Russo's formula.