  We construct pseudorandom generators that fool functions of halfspaces
(threshold functions) under a very broad class of product distributions. This
class includes not only familiar cases such as the uniform distribution on the
discrete cube, the uniform distribution on the solid cube, and the multivariate
Gaussian distribution, but also includes any product of discrete distributions
with probabilities bounded away from 0.
  Our first main result shows that a recent pseudorandom generator construction
of Meka and Zuckerman [MZ09], when suitably modifed, can fool arbitrary
functions of d halfspaces under product distributions where each coordinate has
bounded fourth moment. To eps-fool any size-s, depth-d decision tree of
halfspaces, our pseudorandom generator uses seed length O((d log(ds/eps)+log n)
log(ds/eps)). For monotone functions of d halfspaces, the seed length can be
improved to O((d log(d/eps)+log n) log(d/eps)). We get better bounds for larger
eps; for example, to 1/polylog(n)-fool all monotone functions of (log n)= log
log n halfspaces, our generator requires a seed of length just O(log n). Our
second main result generalizes the work of Diakonikolas et al. [DGJ+09] to show
that bounded independence suffices to fool functions of halfspaces under
product distributions. Assuming each coordinate satisfies a certain stronger
moment condition, we show that any function computable by a size-s, depth-d
decision tree of halfspaces is eps-fooled by O(d^4s^2/eps^2)-wise independence.
