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 poly(n, 2), 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, γ). More generally, given access to t ≤ k different product distributions, the functions may be learned in time npoly(n, 2, γ). Our techniques involve novel results in Fourier analysis relating Fourier expansions with respect to different biases and a generalization of Russo’s formula.