  `Distribution regression' refers to the situation where a response Y depends
on a covariate P where P is a probability distribution. The model is Y=f(P) +
mu where f is an unknown regression function and mu is a random error.
Typically, we do not observe P directly, but rather, we observe a sample from
P. In this paper we develop theory and methods for distribution-free versions
of distribution regression. This means that we do not make distributional
assumptions about the error term mu and covariate P. We prove that when the
effective dimension is small enough (as measured by the doubling dimension),
then the excess prediction risk converges to zero with a polynomial rate.
