Partial AUC Maximization from Positive-unlabeled Data

Maximizing the area under the receiver operating characteristic curve (AUC) is a standard approach to imbalanced binary classification. Although positive and negative data are required for maximizing the AUC, negative data are often difficult to collect in some real-world applications due to privacy concerns or the need for specialized expertise to annotate them. Thus, AUC maximization from positive and unlabeled (PU) data has been attracting attention. Existing methods assume that labeled positive data are unbiased samples from the true positive distribution. However, this ideal assumption is often violated in practice. In this paper, we propose a method to maximize the AUC from biased PU data. To address the bias, our key idea is to exploit {\it confidence}, i.e., the probability that an instance is positive, associated with the small number of labeled positive data. We derive an estimator of the AUC risk using biased PU data with confidence, enabling AUC maximization under such bias. We further show that the rewritten AUC risk induces a Bayes-optimal AUC ranking even when the available confidence is any strictly increasing transformation of the true posterior probability. We experimentally show the effectiveness of our method on eight real-world datasets.

We propose a new method of learning from positive and unlabeled (PU) examples in highly imbalanced datasets. Many real-world problems, such as disease gene identification, targeted marketing, fraud detection, and recommender systems, are hard to address with machine learning methods, due to limited labeled data. Often, training data comprises positive and unlabeled instances, the latter typically being dominated by negative, but including also several positive instances. While PU learning is well-studied, few methods address imbalanced settings or hard-to-detect positive examples that resemble negative ones. Our approach uses a focused empirical risk estimator, incorporating both positive and unlabeled examples to train binary classifiers. Empirical evaluations demonstrate state-of-the-art performance on imbalanced datasets under two labeling mechanisms - selecting positives completely at random (SCAR) and selecting at random (SAR). Beyond these controlled experiments, we demonstrate the value of the proposed method in the real-world application of financial misstatement detection.

Positive-Unlabeled (PU) learning aims to achieve high-accuracy binary classification with limited labeled positive examples and numerous unlabeled ones. Existing cost-sensitive-based methods often rely on strong assumptions that examples with an observed positive label were selected entirely at random. In fact, the uneven distribution of labels is prevalent in real-world PU problems, indicating that most actual positive and unlabeled data are subject to selection bias. Building on the SAR-PU propensity-weighted framework of Bekker et al., we study a PU learning enhancement (PUe) framework using normalized propensity scores and normalized inverse probability weighting (NIPW). PUe's main contributions are a normalized inverse-probability-weighted PU risk formulation; additional theoretical analyses of normalized sample-weight error and common PU estimators under biased labeling; regularized deep propensity-score estimation; integration with modern cost-sensitive PU methods; and support for selectively labeled negative classes. Experiments on MNIST, CIFAR-10, and ADNI demonstrate improvements over several PU baselines under non-uniform label distributions.