This paper studies kernel ridge regression in high dimensions under covariate
shifts and analyzes the role of importance re-weighting. We first derive the
asymptotic expansion of high dimensional kernels under covariate shifts. By a
bias-variance decomposition, we theoretically demonstrate that the re-weighting
strategy allows for decreasing the variance. For bias, we analyze the
regularization of the arbitrary or well-chosen scale, showing that the bias can
behave very differently under different regularization scales. In our analysis,
the bias and variance can be characterized by the spectral decay of a
data-dependent regularized kernel: the original kernel matrix associated with
an additional re-weighting matrix, and thus the re-weighting strategy can be
regarded as a data-dependent regularization for better understanding. Besides,
our analysis provides asymptotic expansion of kernel functions/vectors under
covariate shift, which has its own interest.

本文研究了在协变量转移下高维核岭回归，并分析了重要性重新加权的作用。我们首先推导了协变量转移下高维核的渐近展开式。通过偏差 - 方差分解，我们在理论上证明了重新加权策略可以降低方差。对于偏差，我们分析了任意或精心选择的规范化尺度对偏差的影响，表明偏差在不同的规范化尺度下表现得非常不同。在我们的分析中，偏差和方差可以由一个数据相关的正则化核的谱衰减特性来描述：原始核矩阵与附加的重新加权矩阵相关联，因此重新加权策略可以被视为一种数据相关的正则化以便更好地理解。此外，我们的分析还提供了协变量转移下核函数 / 向量的渐近展开，具有自己的研究兴趣。