  We study the minimax rates of the label shift problem in non-parametric
classification. In addition to the unsupervised setting in which the learner
only has access to unlabeled examples from the target domain, we also consider
the setting in which a small number of labeled examples from the target domain
is available to the learner. Our study reveals a difference in the difficulty
of the label shift problem in the two settings, and we attribute this
difference to the availability of data from the target domain to estimate the
class conditional distributions in the latter setting. We also show that a
class proportion estimation approach is minimax rate-optimal in the
unsupervised setting.
