标题： 在基于RKHS的非参数回归中最优处理协变量偏移 显示英文标题

标题： Optimally tackling covariate shift in RKHS-based nonparametric regression

摘要： 我们研究在再生核希尔伯特空间（RKHS）上的非参数回归中的协变量偏移问题。 我们关注使用源分布和目标分布之间的似然比定义的两个自然的协变量偏移问题族。 当似然比一致有界时，我们证明了具有精心选择正则化参数的核岭回归（KRR）估计量对于具有规则核特征值的大量RKHS来说是最小最大率最优的（除去一个对数因子）。 有趣的是，KRR除了需要似然比的上界外，并不需要完全了解似然比。 与没有协变量偏移的标准统计设置形成鲜明对比的是，我们还证明，在协变量偏移下，一个朴素的估计量，即在函数类上最小化经验风险的估计量，相比于KRR是严格次优的。 然后，我们处理更大的协变量偏移问题类，其中似然比可能无界，但具有有限的二阶矩。 在此，我们提出了一种加权KRR估计量，该估计量基于对似然比的仔细截断来加权样本。 同样，我们能够证明该估计量是最小最大率最优的，除去对数因子。 显示英文摘要

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 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.