Optimally tackling covariate shift in RKHS-based nonparametric regression
This addresses covariate shift problems in machine learning for researchers, providing theoretical guarantees but is incremental as it builds on existing RKHS methods.
The paper tackles covariate shift in nonparametric regression by analyzing kernel ridge regression (KRR) and proposing a reweighted estimator, proving minimax rate-optimality (up to log factors) for bounded and unbounded likelihood ratios.
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 likelihood ratios apart from an upper bound on them. 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 sub-optimal under covariate shift as compared to KRR. We then address the larger class of covariate shift problems where the 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 rate-optimal, up to logarithmic factors.