A Variational Approach for Learning from Positive and Unlabeled Data

Learning binary classifiers only from positive and unlabeled (PU) data is an important and challenging task in many real-world applications, including web text classification, disease gene identification and fraud detection, where negative samples are difficult to verify experimentally. Most recent PU learning methods are developed based on the conventional misclassification risk of the supervised learning type, and they require to solve the intractable risk estimation problem by approximating the negative data distribution or the class prior. In this paper, we introduce a variational principle for PU learning that allows us to quantitatively evaluate the modeling error of the Bayesian classifier directly from given data. This leads to a loss function which can be efficiently calculated without any intermediate step or model, and a variational learning method can then be employed to optimize the classifier under general conditions. In addition, the discriminative performance and numerical stability of the variational PU learning method can be further improved by incorporating a margin maximizing loss function. We illustrate the effectiveness of the proposed variational method on a number of benchmark examples.