  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.
