## Classification with a Reject Option using a Hinge Loss

** Peter L. Bartlett, Marten H. Wegkamp**; 9(59):1823−1840, 2008.

### Abstract

We consider the problem of binary classification where the
classifier can, for a particular cost, choose not to classify an
observation. Just as in the conventional classification problem,
minimization of the sample average of the cost is a difficult
optimization problem. As an alternative, we propose the optimization
of a certain convex loss function φ, analogous to the hinge
loss used in support vector machines (SVMs). Its convexity ensures
that the sample average of this surrogate loss can be efficiently
minimized. We study its statistical properties. We show that
minimizing the expected surrogate loss—the φ-risk—also
minimizes the risk. We also study the rate at which the φ-risk
approaches its minimum value. We show that fast rates are possible
when the conditional probability *P*(*Y*=1|*X*) is unlikely to be
close to certain critical values.

© JMLR 2008. (edit, beta) |