Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex.

Area under ROC (AUC) is a metric which is widely used for measuring the classification performance for imbalanced data.

Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex.

Nonetheless, most of the existing methods could only optimize P AUC approximately, leading to inevitable biases that are not controllable.

We collect in Table A.1 the notations of performance measures used in this paper.X input space Y output space Z sample space S training dataset n sample size z To this aim, we require the following lemma on the self-bounding property of smooth loss functions. We only consider Part (b). We can plug the above inequality back into (B.1), and get E[F ( A (S)) F In this section, we prove Theorem 2. To this aim, we first introduce some lemmas. Lemma C.3 is motivated by a recent Let p 2 be any number. C.1 to show that 1 null The stated bound then follows by combining the above two inequalities together.

Maximizing the area under the receiver operating characteristic curve (AUC) is a popular approach to imbalanced binary classification problems. Existing AUC maximization methods usually assume that training and test distributions are identical. However, this assumption is often violated in practice due to {\it a positive distribution shift}, where the negative-conditional density does not change but the positive-conditional density can vary. This shift often occurs in imbalanced classification since positive data are often more diverse and time-varying than negative data. To deal with this shift, we theoretically show that the AUC on the test distribution can be expressed by using the positive and marginal training densities and the marginal test density.