Improved Bounds for Learning Symmetric Juntas

The problem was first proposed by Blum [1] and Blum and Langley [2]. Ever since, the first nontrivial algorithm was given by [4], which runs in time n0.7kpoly(log 1/δ, 2k, n), for general juntas and n 2 3 poly(log 1/δ, 2k, n) for symmetric juntas. We give an algorithm for symmetric juntas which runs in time nk/3(1+o(1))poly(log 1/δ, 2k, n). We further show that when k is bigger than some large enough constant, the algorithm runs in time n0.18kpoly(log 1/δ, 2k, n). To our knowledge, this is the best known upper bound for learning symmetric juntas under the uniform distribution. The same algorithm has also been proposed by [5]. In [5] it was shown that the running time is bounded by nk/2(1+o(1))poly(log 1/δ, 2k, n). It was also shown that under a certain number theoretic assumption, the running time is nk/3poly(log 1/δ, 2k, n).