In credit fraud detection practice, certain fraudulent transactions often evade detection because of the hidden nature of fraudulent behavior. To address this issue, an increasing number of positive-unlabeled (PU) learning techniques have been employed by more and more financial institutions. However, most of these methods are designed for single data sets and do not take into account the heterogeneity of data when they are collected from different sources. In this paper, we propose an integrative PU learning method (I-PU) for pooling information from multiple heterogeneous PU data sets. A novel approach that penalizes group differences is developed to explicitly and automatically identify the cluster structures of coefficients across different data sets, thus offering a plausible interpretation of heterogeneity. Furthermore, we apply a bilevel selection method to detect the sparse structure at both the group level and within-group level. Theoretically, we show that our proposed estimator has the oracle property. Computationally, we design an expectation-maximization (EM) algorithm framework and propose an alternating direction method of multipliers (ADMM) algorithm to solve it. Simulation results show that our proposed method has better numerical performance in terms of variable selection, parameter estimation, and prediction ability. Finally, a real-world application showcases the effectiveness of our method in identifying distinct coefficient clusters and its superior prediction performance compared with direct data merging or separate modeling. This result also offers valuable insights for financial institutions in developing targeted fraud detection systems. History: Accepted by Ram Ramesh, Area Editor for Data Science & Machine Learning. Funding: This work was supported by the National Natural Science Foundation of China [Grants 72071169, 72231005, 72233002, and 72471169], the Fundamental Research Funds for the Central Universities of China [Grant 20720231060], the National Social Science Fund of China [Grant 21&ZD146], and Shuimu Tsinghua Scholar Program. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0366 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0366 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

In diverse fields ranging from finance to omics, it is increasingly common that data is distributed with multiple individual sources (referred to as “clients” in some studies). Integrating raw data, although powerful, is often not feasible, for example, when there are considerations on privacy protection. Distributed learning techniques have been developed to integrate summary statistics as opposed to raw data. In many existing distributed learning studies, it is stringently assumed that all the clients have the same model. To accommodate data heterogeneity, some federated learning methods allow for client-specific models. In this article, we consider the scenario that clients form clusters, those in the same cluster have the same model, and different clusters have different models. Further considering the clustering structure can lead to a better understanding of the “interconnections” among clients and reduce the number of parameters. To this end, we develop a novel penalization approach. Specifically, group penalization is imposed for regularized estimation and selection of important variables, and fusion penalization is imposed to automatically cluster clients. An effective ADMM algorithm is developed, and the estimation, selection, and clustering consistency properties are established under mild conditions. Simulation and data analysis further demonstrate the practical utility and superiority of the proposed approach.