  In a recent breakthrough, [Bshouty et al., 2005] obtained the first
passive-learning algorithm for DNFs under the uniform distribution. They showed
that DNFs are learnable in the Random Walk and Noise Sensitivity models. We
extend their results in several directions. We first show that thresholds of
parities, a natural class encompassing DNFs, cannot be learned efficiently in
the Noise Sensitivity model using only statistical queries. In contrast, we
show that a cyclic version of the Random Walk model allows to learn efficiently
polynomially weighted thresholds of parities. We also extend the algorithm of
Bshouty et al. to the case of Unions of Rectangles, a natural generalization of
DNFs to $\{0,...,b-1\}^n$.
