We study covariate shift in the context of nonparametric regression. We introduce a new measure of distribution mismatch between the source and target distributions that is based on the integrated ratio of probabilities of balls at a given radius. We use the scaling of this measure with respect to the radius to characterize the minimax rate of estimation over a family of H\"older continuous functions under covariate shift. In comparison to the recently proposed notion of transfer exponent, this measure leads to a sharper rate of convergence and is more fine-grained. We accompany our theory with concrete instances of covariate shift that illustrate this sharp difference.
翻译:我们从非参数回归的角度研究非参数回归情况下的共变式变化。 我们引入了一种基于某个半径球概率综合比的源与目标分布不匹配的新度量。 我们使用对半径的这一量度的缩放来描述一个H\"老的连续函数家族在共变式变化下的最小估计速率。 与最近提出的转移指数概念相比, 这一度量导致更显著的趋同率, 并且更精细的加分。 我们用共变变化的具体例子来说明这一显著差异。