  In supervised learning, we often face with ambiguous (A) samples that are
difficult to label even by domain experts. In this paper, we consider a binary
classification problem in the presence of such A samples. This problem is
substantially different from semi-supervised learning since unlabeled samples
are not necessarily difficult samples. Also, it is different from 3-class
classification with the positive (P), negative (N), and A classes since we do
not want to classify test samples into the A class. Our proposed method extends
binary classification with reject option, which trains a classifier and a
rejector simultaneously using P and N samples based on the 0-1-$c$ loss with
rejection cost $c$. More specifically, we propose to train a classifier and a
rejector under the 0-1-$c$-$d$ loss using P, N, and A samples, where $d$ is the
misclassification penalty for ambiguous samples. In our practical
implementation, we use a convex upper bound of the 0-1-$c$-$d$ loss for
computational tractability. Numerical experiments demonstrate that our method
can successfully utilize the additional information brought by such A training
data.
