  We study the domain adaptation problem with label shift in this work. Under
the label shift context, the marginal distribution of the label varies across
the training and testing datasets, while the conditional distribution of
features given the label is the same. Traditional label shift adaptation
methods either suffer from large estimation errors or require cumbersome
post-prediction calibrations. To address these issues, we first propose a
moment-matching framework for adapting the label shift based on the geometry of
the influence function. Under such a framework, we propose a novel method named
\underline{E}fficient \underline{L}abel \underline{S}hift
\underline{A}daptation (ELSA), in which the adaptation weights can be estimated
by solving linear systems. Theoretically, the ELSA estimator is
$\sqrt{n}$-consistent ($n$ is the sample size of the source data) and
asymptotically normal. Empirically, we show that ELSA can achieve
state-of-the-art estimation performances without post-prediction calibrations,
thus, gaining computational efficiency.
