1 research outputs found
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 the
hypercube with polynomially small error, the best construction known requires
seed-length O(log^2 n) (MekaZ13). 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:
1) We give a near optimal derandomization of the Chernoff bound for
independent, uniformly random bits. Specifically, we show how to generate a x
in {1,-1}^n using random bits such that for any
unit vector u, matches the sub-Gaussian tail behaviour predicted by the
Chernoff bound up to error eps.
2) We construct a generator which fools halfspaces with {0,1,-1} coefficients
with error eps with a seed-length of . This
includes the important special case of majorities.
In both cases, the best previous results required seed-length of .
Technically, our work combines new Fourier-analytic tools with the iterative
dimension reduction techniques and the gradually increasing independence
paradigm of previous works (KaneMN11, CelisRSW13, GopalanMRTV12)