  A Forster transform is an operation that turns a distribution into one with
good anti-concentration properties. While a Forster transform does not always
exist, we show that any distribution can be efficiently decomposed as a
disjoint mixture of few distributions for which a Forster transform exists and
can be computed efficiently. As the main application of this result, we obtain
the first polynomial-time algorithm for distribution-independent PAC learning
of halfspaces in the Massart noise model with strongly polynomial sample
complexity, i.e., independent of the bit complexity of the examples. Previous
algorithms for this learning problem incurred sample complexity scaling
polynomially with the bit complexity, even though such a dependence is not
information-theoretically necessary.
