Making an erroneous decision may cause serious results in diverse mission-critical tasks such as medical diagnosis and bioinformatics. Previous work focuses on classification with a reject option, i.e., abstain rather than classify an instance of low confidence. Most mission-critical tasks are always accompanied with class imbalance and cost sensitivity, where AUC has been shown a preferable measure than accuracy in classification. In this work, we propose the framework of AUC optimization with a reject option, and the basic idea is to withhold the decision of ranking a pair of positive and negative instances with a lower cost, rather than mis-ranking. We obtain the Bayes optimal solution for ranking, and learn the reject function and score function for ranking, simultaneously. An online algorithm has been developed for AUC optimization with a reject option, by considering the convex relaxation and plug-in rule. We verify, both theoretically and empirically, the effectiveness of the proposed algorithm.

Decision trees have attracted much attention during the past decades. Previous decision trees include axis-parallel and oblique decision trees; both of them try to find the best splits via exhaustive search or heuristic algorithms in each iteration. Oblique decision trees generally simplify tree structure and take better performance, but are always accompanied with higher computation, as well as the initialization with the best axis-parallel splits. This work presents the Weighted Oblique Decision Tree (WODT) based on continuous optimization with random initialization. We consider different weights of each instance for child nodes at all internal nodes, and then obtain a split by optimizing the continuous and differentiable objective function of weighted information entropy. Extensive experiments show the effectiveness of the proposed algorithm.