Learning from Positive and Unlabeled Data with Arbitrary Positive Shift

Positive-unlabeled (PU) learning trains a binary classifier using only positive and unlabeled data. A common simplifying assumption is that the positive data is representative of the target positive class. This assumption rarely holds in practice due to temporal drift, domain shift, and/or adversarial manipulation. This paper shows that PU learning is possible even with arbitrarily non-representative positive data given unlabeled data from the source and target distributions. Our key insight is that only the negative class's distribution need be fixed. We integrate this into two statistically consistent methods to address arbitrary positive bias -one approach combines negative-unlabeled learning with unlabeled-unlabeled learning while the other uses a novel, recursive risk estimator. Experimental results demonstrate our methods' effectiveness across numerous real-world datasets and forms of positive bias, including disjoint positive class-conditional supports. Additionally, we propose a general, simplified approach to address PU risk estimation overfitting. Introduction Positive-negative (PN) learning (i.e., ordinary supervised classification) trains a binary classifier using positive and negative labeled datasets. In practice, good labeled data are often unavailable for one class. High negative-class diversity may make constructing a representative labeled set prohibitively difficult [1], or negative data may not be systematically recorded in some domains [2] . Positive-unlabeled (PU) learning addresses this problem by constructing classifiers using only labeled-positive and unlabeled data. PU learning has been applied to numerous real-world domains including: opinion spam detection [3], disease-gene identification [4], land-cover classification [5], and protein similarity prediction [6] . The related task of negative-unlabeled (NU) learning is functionally identical to PU learning but with labeled data drawn from the negative class. Most PU learning methods assume the labeled set is selected completely at random (SCAR) from the target distribution [1, 6, 7, 8, 9, 10, 11] . External factors like temporal drift, domain shift, and adversarial concept drift often cause the labeled-positive and target distributions to diverge. Biased-positive, unlabeled (bPU) learning algorithms relax SCAR by modeling sample selection bias for the labeled data [12, 13] or a covariate shift between the training and target distributions [14]. This paper generalizes bPU learning to the more challenging arbitrary-positive, unlabeled (aPU) learning setting, where the labeled (positive) data may be arbitrarily different from the target distribution's positive class. Solving this problem would eliminate the need to spend time and money labeling new data whenever the positive class drifts. 34th Conference on Neural Information Processing Systems (NeurIPS 2020),