  In most classification tasks there are observations that are ambiguous and
therefore difficult to correctly label. Set-valued classifiers output sets of
plausible labels rather than a single label, thereby giving a more appropriate
and informative treatment to the labeling of ambiguous instances. We introduce
a framework for multiclass set-valued classification, where the classifiers
guarantee user-defined levels of coverage or confidence (the probability that
the true label is contained in the set) while minimizing the ambiguity (the
expected size of the output). We first derive oracle classifiers assuming the
true distribution to be known. We show that the oracle classifiers are obtained
from level sets of the functions that define the conditional probability of
each class. Then we develop estimators with good asymptotic and finite sample
properties. The proposed estimators build on existing single-label classifiers.
The optimal classifier can sometimes output the empty set, but we provide two
solutions to fix this issue that are suitable for various practical needs.
