Online Adaptation to Class-Prior Shift from Positive and Unlabeled Data

Class-prior shift is a common distribution shift in which the class-conditional distributions remain stable while the class proportions change. Existing online label-shift methods assume labeled source data from every class. We ask: what is the statistical price of online adaptation when the source supervision itself is incomplete? We study this question when the source data contain only positive and unlabeled (PU) examples, while unlabeled target observations arrive sequentially and the class prior changes over time. Such data arise, for example, in protein screening, where only sequences that pass the screen are labeled. We show that labeled negatives are not needed to preserve the online tracking rate. Under a source-sample stability condition, the temporal component of the cumulative squared prior-tracking error retains the $\tilde{\mathcal{O}}(T^{1/3}{V}_T^{2/3})$ dependence on the horizon $T$ and total variation $V_T$ achieved with positive and negative source data. The effect of PU supervision appears in the finite-source term: reconstructing the unobserved negative class amplifies source-estimation errors with weights that depend on where the target-prior path lies, beyond $V_T$ alone. We further derive a dynamic classification-regret bound for online threshold adaptation against the best threshold of the same fixed score at each target prior. Experiments on image and text benchmarks show that threshold adaptation improves classification under substantial prior shift with low online cost without retraining the initial model.