  The problem of statistical learning is to construct a predictor of a random
variable $Y$ as a function of a related random variable $X$ on the basis of an
i.i.d. training sample from the joint distribution of $(X,Y)$. Allowable
predictors are drawn from some specified class, and the goal is to approach
asymptotically the performance (expected loss) of the best predictor in the
class. We consider the setting in which one has perfect observation of the
$X$-part of the sample, while the $Y$-part has to be communicated at some
finite bit rate. The encoding of the $Y$-values is allowed to depend on the
$X$-values. Under suitable regularity conditions on the admissible predictors,
the underlying family of probability distributions and the loss function, we
give an information-theoretic characterization of achievable predictor
performance in terms of conditional distortion-rate functions. The ideas are
illustrated on the example of nonparametric regression in Gaussian noise.
