  As a fundamental problem in machine learning, dataset shift induces a
paradigm to learn and transfer knowledge under changing environment. Previous
methods assume the changes are induced by covariate, which is less practical
for complex real-world data. We consider the Generalized Label Shift (GLS),
which provides an interpretable insight into the learning and transfer of
desirable knowledge. Current GLS methods: 1) are not well-connected with the
statistical learning theory; 2) usually assume the shifting conditional
distributions will be matched with an implicit transformation, but its explicit
modeling is unexplored. In this paper, we propose a conditional adaptation
framework to deal with these challenges. From the perspective of learning
theory, we prove that the generalization error of conditional adaptation is
lower than previous covariate adaptation. Following the theoretical results, we
propose the minimum uncertainty principle to learn conditional invariant
transformation via discrepancy optimization. Specifically, we propose the
\textit{conditional metric operator} on Hilbert space to characterize the
distinctness of conditional distributions. For finite observations, we prove
that the empirical estimation is always well-defined and will converge to
underlying truth as sample size increases. The results of extensive experiments
demonstrate that the proposed model achieves competitive performance under
different GLS scenarios.
