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

Class-prior shift is a common form of distribution shift in which the class-conditional distributions remain stable while the class proportions change. Existing online label-shift methods typically assume labeled offline data from every class. We study a weaker setting in which the offline source data contain only labeled positive examples and unlabeled examples, while unlabeled target observations arrive sequentially during online adaptation and the class prior changes over time. This raises a natural question: \emph{what is the statistical price of online adaptation when labeled negative source examples are unavailable?} Our main finding is that the two sources of difficulty can be separated. Labeled negative examples are not needed to preserve the basic online tracking rate: up to logarithmic and conditioning factors, the cumulative squared prior-tracking error has the same $\widetilde{\mathcal O}(T^{1/3}V_T^{2/3})$ dependence on the horizon $T$ and the total variation $V_T$ as in online label shift with labeled offline data from both classes. PU supervision instead introduces an additional finite-source cost. This cost is path dependent: it depends on where the target-prior sequence lies relative to the source prior and is not determined by $V_T$ alone. Separate lower bounds show that the temporal variation cost and the finite-source PU cost are unavoidable in their respective statistical settings. We further derive a dynamic classification regret bound for online threshold adaptation while keeping the underlying classification score fixed. The method therefore adapts the prior and decision threshold online without retraining the initial model. Preliminary synthetic and digits experiments illustrate the finite-source effect of PU supervision and show that adaptive thresholding can substantially improve over a static source threshold under class-prior shift.