  In supervised learning, we typically leverage a fully labeled dataset to
design methods for function estimation or prediction. In many practical
situations, we are able to obtain alternative feedback, possibly at a low cost.
A broad goal is to understand the usefulness of, and to design algorithms to
exploit, this alternative feedback. In this paper, we consider a
semi-supervised regression setting, where we obtain additional ordinal (or
comparison) information for the unlabeled samples. We consider ordinal feedback
of varying qualities where we have either a perfect ordering of the samples, a
noisy ordering of the samples or noisy pairwise comparisons between the
samples. We provide a precise quantification of the usefulness of these types
of ordinal feedback in both nonparametric and linear regression, showing that
in many cases it is possible to accurately estimate an underlying function with
a very small labeled set, effectively \emph{escaping the curse of
dimensionality}. We also present lower bounds, that establish fundamental
limits for the task and show that our algorithms are optimal in a variety of
settings. Finally, we present extensive experiments on new datasets that
demonstrate the efficacy and practicality of our algorithms and investigate
their robustness to various sources of noise and model misspecification.
