  We present new minimax results that concisely capture the relative benefits
of source and target labeled data, under covariate-shift. Namely, we show that
the benefits of target labels are controlled by a transfer-exponent $\gamma$
that encodes how singular Q is locally w.r.t. P, and interestingly allows
situations where transfer did not seem possible under previous insights. In
fact, our new minimax analysis - in terms of $\gamma$ - reveals a continuum of
regimes ranging from situations where target labels have little benefit, to
regimes where target labels dramatically improve classification. We then show
that a recently proposed semi-supervised procedure can be extended to adapt to
unknown $\gamma$, and therefore requests labels only when beneficial, while
achieving minimax transfer rates.
