OPTIMALLY TACKLING COVARIATE SHIFT IN RKHS-BASED NONPARAMETRIC REGRESSION
成果类型：
Article
署名作者：
Ma, Cong; Pathak, Reese; Wainwright, Martin J.
署名单位：
University of Chicago; University of California System; University of California Berkeley; Massachusetts Institute of Technology (MIT)
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/23-AOS2268
发表日期：
2023
页码：
738-761
关键词：
摘要：
We study the covariate shift problem in the context of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We focus on two natural families of covariate shift problems defined using the likelihood ratios between the source and target distributions. When the likelihood ratios are uniformly bounded, we prove that the kernel ridge regression (KRR) estimator with a carefully chosen regularization parameter is minimax rate-optimal (up to a log factor) for a large family of RKHSs with regular kernel eigenvalues. Interestingly, KRR does not require full knowledge of the likelihood ratio apart from an upper bound on it. In striking contrast to the standard statistical setting without covariate shift, we also demonstrate that a naive estimator, which minimizes the empirical risk over the function class, is strictly suboptimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where likelihood ratio is possibly unbounded yet has a finite second moment. Here, we propose a reweighted KRR estimator that weights samples based on a careful truncation of the likelihood ratios. Again, we are able to show that this estimator is minimax optimal, up to logarithmic factors.