Active learning from positive and unlabeled examples
Farnam Mansouri, Sandra Zilles, Shai Ben-David
TL;DR
This work analyzes active learning for binary classification from positive and unlabeled data under the SCAR assumption, focusing on label complexity when label feedback is stochastic with probability $\omega$. It develops a disagreement-based approach, adapting the CAL algorithm to a PU setting and, for the known-$\pi_{\mathcal{D}}$ case, proves that the disagreement mass can be halved after $O(d\theta)$ labeled queries, yielding a bound that scales as $O\left( \frac{\ln(1/\varepsilon)\,\theta^2\bigl(d\ln(\theta) + \ln\ln(1/\varepsilon) + \ln(1/\delta)\bigr)}{\omega} \right)$. For unknown $\pi_{\mathcal{D}}$, it introduces EstRate and a binary-search over the estimated positive rate to progressively constrain the version space, resulting in a two-term label complexity bound that combines a $\omega$-dependent term with a $\pi_{\mathcal{D}}$-dependent term. Overall, the paper provides the first formal guarantees for active PU learning, linking label efficiency to the disagreement coefficient $\theta$, the VC dimension $d$, and the feedback rate $\omega$, with practical implications for applications like anomaly detection and recommender systems. The results also highlight open questions on improving the $\theta^2$ dependence and extending to discrete or agnostic settings.
Abstract
Learning from positive and unlabeled data (PU learning) is a weakly supervised variant of binary classification in which the learner receives labels only for (some) positively labeled instances, while all other examples remain unlabeled. Motivated by applications such as advertising and anomaly detection, we study an active PU learning setting where the learner can adaptively query instances from an unlabeled pool, but a queried label is revealed only when the instance is positive and an independent coin flip succeeds; otherwise the learner receives no information. In this paper, we provide the first theoretical analysis of the label complexity of active PU learning.