  We consider the problem of cost sensitive multiclass classification, where we
would like to increase the sensitivity of an important class at the expense of
a less important one. We adopt an {\em apportioned margin} framework to address
this problem, which enables an efficient margin shift between classes that
share the same boundary. The decision boundary between all pairs of classes
divides the margin between them in accordance to a given prioritization vector,
which yields a tighter error bound for the important classes while also
reducing the overall out-of-sample error. In addition to demonstrating an
efficient implementation of our framework, we derive generalization bounds,
demonstrate Fisher consistency, adapt the framework to Mercer's kernel and to
neural networks, and report promising empirical results on all accounts.
