Positive-Unlabeled (PU) learning presents unique challenges due to the lack of explicitly labeled negative samples, particularly in high-stakes domains such as fraud detection and medical diagnosis. To address data scarcity and privacy constraints, we propose a novel transfer learning with model averaging framework that integrates information from heterogeneous data sources - including fully binary labeled, semi-supervised, and PU data sets - without direct data sharing. For each source domain type, a tailored logistic regression model is conducted, and knowledge is transferred to the PU target domain through model averaging. Optimal weights for combining source models are determined via a cross-validation criterion that minimizes the Kullback-Leibler divergence. We establish theoretical guarantees for weight optimality and convergence, covering both misspecified and correctly specified target models, with further extensions to high-dimensional settings using sparsity-penalized estimators. Extensive simulations and real-world credit risk data analyses demonstrate that our method outperforms other comparative methods in terms of predictive accuracy and robustness, especially under limited labeled data and heterogeneous environments.

We study estimation of class prior for unlabeled target samples which is possibly different from that of source population. It is assumed that for the source data only samples from positive class and from the whole population are available (PU learning scenario). We introduce a novel direct estimator of class prior which avoids estimation of posterior probabilities and has a simple geometric interpretation. It is based on a distribution matching technique together with kernel embedding and is obtained as an explicit solution to an optimisation task. We establish its asymptotic consistency as well as a non-asymptotic bound on its deviation from the unknown prior, which is calculable in practice. We study finite sample behaviour for synthetic and real data and show that the proposal, together with a suitably modified version for large values of source prior, works on par or better than its competitors.

The basic problem studied in the paper concerns learning from data which is only partially labelled, but nonetheless doing better than with fully labelled data. In the particular scenario where one has a pool of unlabelled data in lieu of one of the classes, the paper seeks to quantify the impact this has on the estimation error of the learned classifier. Determining the degradation or lack thereof when learning from unlabelled data is interesting, and thus the paper seems well motivated. The machinery used to illustrate its messages are fairly standard -- the key quantities, namely the estimation errors for each scenario, are derived from a simple Rademacher analysis -- however, the final results appear novel, with implications worked through in various scenarios. The papers hinges on the simple facts that (a) different risks (for PN/PU/NU learning) may be seen as employing different weightings on individual risks for the positive and negative class, and (b) these ratings are reflected in appropriate terms for the estimation error.

This study introduces a new approach to addressing positive and unlabeled (PU) data through the double exponential tilting model (DETM). Traditional methods often fall short because they only apply to selected completely at random (SCAR) PU data, where the labeled positive and unlabeled positive data are assumed to be from the same distribution. In contrast, our DETM's dual structure effectively accommodates the more complex and underexplored selected at random PU data, where the labeled and unlabeled positive data can be from different distributions. We rigorously establish the theoretical foundations of DETM, including identifiability, parameter estimation, and asymptotic properties. Additionally, we move forward to statistical inference by developing a goodness-of-fit test for the SCAR condition and constructing confidence intervals for the proportion of positive instances in the target domain. We leverage an approximated Bayes classifier for classification tasks, demonstrating DETM's robust performance in prediction. Through theoretical insights and practical applications, this study highlights DETM as a comprehensive framework for addressing the challenges of PU data.

We propose a meta-learning method for positive and unlabeled (PU) classification, which improves the performance of binary classifiers obtained from only PU data in unseen target tasks. PU learning is an important problem since PU data naturally arise in real-world applications such as outlier detection and information retrieval. Existing PU learning methods require many PU data, but sufficient data are often unavailable in practice. The proposed method minimizes the test classification risk after the model is adapted to PU data by using related tasks that consist of positive, negative, and unlabeled data. We formulate the adaptation as an estimation problem of the Bayes optimal classifier, which is an optimal classifier to minimize the classification risk. The proposed method embeds each instance into a task-specific space using neural networks. With the embedded PU data, the Bayes optimal classifier is estimated through density-ratio estimation of PU densities, whose solution is obtained as a closed-form solution. The closed-form solution enables us to efficiently and effectively minimize the test classification risk. We empirically show that the proposed method outperforms existing methods with one synthetic and three real-world datasets.

Positive-unlabeled (PU) learning deals with binary classification problems when only positive (P) and unlabeled (U) data are available. Many recent PU methods are based on neural networks, but little has been done to develop boosting algorithms for PU learning, despite boosting algorithms' strong performance on many fully supervised classification problems. In this paper, we propose a novel boosting algorithm, AdaPU, for PU learning. Similarly to AdaBoost, AdaPU aims to optimize an empirical exponential loss, but the loss is based on the PU data, rather than on positive-negative (PN) data. As in AdaBoost, we learn a weighted combination of weak classifiers by learning one weak classifier and its weight at a time. However, AdaPU requires a very different algorithm for learning the weak classifiers and determining their weights. This is because AdaPU learns a weak classifier and its weight using a weighted positive-negative (PN) dataset with some negative data weights $-$ the dataset is derived from the original PU data, and the data weights are determined by the current weighted classifier combination, but some data weights are negative. Our experiments showed that AdaPU outperforms neural networks on several benchmark PU datasets, including a large-scale challenging cyber security dataset.

Learning from positive and unlabeled data or PU learning is the setting where a learner only has access to positive examples and unlabeled data. The assumption is that the unlabeled data can contain both positive and negative examples. This setting has attracted increasing interest within the machine learning literature as this type of data naturally arises in applications such as medical diagnosis and knowledge base completion. This article provides a survey of the current state of the art in PU learning. It proposes seven key research questions that commonly arise in this field and provides a broad overview of how the field has tried to address them.

Most positive and unlabeled data is subject to selection biases. The labeled examples can, for example, be selected from the positive set because they are easier to obtain or more obviously positive. This paper investigates how learning can be enabled in this setting. We propose and theoretically analyze an empirical-risk-based method for incorporating the labeling mechanism. Additionally, we investigate under which assumptions learning is possible when the labeling mechanism is not fully understood and propose a practical method to enable this. Our empirical analysis supports the theoretical results and shows that taking into account the possibility of a selection bias, even when the labeling mechanism is unknown, improves the trained classifiers.

When learning from positive and unlabelled data, it is a strong assumption that the positive observations are randomly sampled from the distribution of $X$ conditional on $Y = 1$, where X stands for the feature and Y the label. Most existing algorithms are optimally designed under the assumption. However, for many real-world applications, the observed positive examples are dependent on the conditional probability $P(Y = 1|X)$ and should be sampled biasedly. In this paper, we assume that a positive example with a higher $P(Y = 1|X)$ is more likely to be labelled and propose a probabilistic-gap based PU learning algorithms. Specifically, by treating the unlabelled data as noisy negative examples, we could automatically label a group positive and negative examples whose labels are identical to the ones assigned by a Bayesian optimal classifier with a consistency guarantee. The relabelled examples have a biased domain, which is remedied by the kernel mean matching technique. The proposed algorithm is model-free and thus do not have any parameters to tune. Experimental results demonstrate that our method works well on both generated and real-world datasets.

In real-world applications, it is conceivable that only positive and unlabeled (PU) data are available for training a classifier. For instance, in land-cover image classification, images of urban regions can be easily labeled, while images of non-urban regions are difficult to annotate due to high diversity of non-urban regions containing, e.g., forest, seas, grasses, and soil (Li et al., 2011). To cope with such situations, PU classification has been actively studied (Letouzey et al., 2000; Elkan and Noto, 2008; du Plessis et al., 2015), and the state-of-the-art method allows us to systematically train deep neural networks only from PU data (Kiryo et al., 2017). However, existing PU classification methods typically require an estimate of the class-prior probability, and their performance is sensitive to the quality of class-prior estimation (Kiryo et al., 2017). Although various class-prior estimation methods from PU data have been proposed so far (du Plessis and Sugiyama, 2014; Ramaswamy et al., 2016; Jain et al., 2016; du Plessis et al., 2017; Northcutt et al., 2017), accurate estimation of the class-prior is still highly challenging particularly for high-dimensional data.