  Many binary classification problems minimize misclassification above (or
below) a threshold. We show that instances of ranking problems, accuracy at the
top or hypothesis testing may be written in this form. We propose a general
framework to handle these classes of problems and show which known methods
(both known and newly proposed) fall into this framework. We provide a
theoretical analysis of this framework and mention selected possible pitfalls
the methods may encounter. We suggest several numerical improvements including
the implicit derivative and stochastic gradient descent. We provide an
extensive numerical study. Based both on the theoretical properties and
numerical experiments, we conclude the paper by suggesting which method should
be used in which situation.
