Pseudorandomness for concentration bounds and signed majorities

The problem of constructing pseudorandom generators that fool halfspaces has been studied intensively in recent times. For fooling halfspaces over {±1}n with polynomially small error, the best construction known requires seed-length O(log(n)) [MZ13]. Getting the seed-length down to O(log(n)) is a natural challenge in its own right, which needs to be overcome in order to derandomize RL. In this work we make progress towards this goal by obtaining near-optimal generators for two important special cases: • We give a near optimal derandomization of the Chernoff bound for independent, uniformly random bits. Specifically, we show how to generate x ∈ {±1}n using Õ(log(n/ε)) random bits such that for any unit vector u, u · x matches the subGaussian tail behaviour predicted by the Chernoff bound up to error ε. • We construct a generator which fools halfspaces with {0, 1,−1} coefficients with error ǫ with a seed-length of Õ(log(n/ǫ)). This includes the important special case of majorities. In both cases, the best previous results required seed-length of O(log n+log(1/ǫ)). Technically, our work combines new Fourier-analytic tools with the iterative dimension reduction techniques and the gradually increasing independence paradigm of previous works [KMN11, CRSW13, GMR12].