arXiv subjects
Publications and source records attributed to Jan Arpe.
We prove that the class of functions g:{-1,+1}^n -> {-1,+1} that only depend on an unknown subset of k< 0 and access to a random walk on {-1,+1}^n labeled by an arbitrary function f:{-1,+1}^n -> {-1,+1}, finds with probability at least 1-delta a k-junta that is (opt(f)+epsilon)-close to f, where opt(f) denotes the distance of a closest k-junta to f.
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 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/t} * poly(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.