Marginal Singularity, and the Benefits of Labels in Covariate-Shift

We present new minimax results that concisely capture the relative benefits of source and target labeled data, under {covariate-shift}. Namely, we show that, in general classification settings, the benefits of target labels are controlled by a \emph{transfer-exponent} $\gamma$ that encodes how \emph{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 \emph{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 target labels only when beneficial, while achieving nearly minimax transfer rates.