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.

我们提出了新的极小极大结果，用于简洁地捕捉协变量偏移下源标记数据和目标标记数据之间的相对收益。我们展示了目标标记的收益由一个传输指数$\gamma$控制，这个指数编码了Q在局部相对于P的奇异性，并且允许了在之前的研究中似乎不可能进行的情况。实际上，我们的新的极小极大分析显示了一个连续的区域，从目标标记几乎没有收益的情况到目标标记显著提高分类的情况。然后，我们展示了最近提出的半监督程序可以扩展到适应未知的$\gamma$，因此只在有效时才请求标签，同时实现最小极大传输速率。

边缘奇点与标签在协变量转移中的优势