  This paper focuses on supervised and unsupervised online label shift, where
the class marginals $Q(y)$ varies but the class-conditionals $Q(x|y)$ remain
invariant. In the unsupervised setting, our goal is to adapt a learner, trained
on some offline labeled data, to changing label distributions given unlabeled
online data. In the supervised setting, we must both learn a classifier and
adapt to the dynamically evolving class marginals given only labeled online
data. We develop novel algorithms that reduce the adaptation problem to online
regression and guarantee optimal dynamic regret without any prior knowledge of
the extent of drift in the label distribution. Our solution is based on
bootstrapping the estimates of \emph{online regression oracles} that track the
drifting proportions. Experiments across numerous simulated and real-world
online label shift scenarios demonstrate the superior performance of our
proposed approaches, often achieving 1-3\% improvement in accuracy while being
sample and computationally efficient. Code is publicly available at
https://github.com/acmi-lab/OnlineLabelShift.
