The supplemental material is organized as follows. Section A provides the results of all the additional synthetic experiments and real data results for various kernel-based methods and the detailed settings. Section B describes the algorithm details we use in Section A. In Section C, we provide some useful lemmas and all the technical proofs of the theoretical results in the main text. In this section, we provide more experiment results, including KRR (Section A.1), KQR for various Section A.7. A.1 Kernel ridge regression For the squared loss, we consider the following two examples. TIRW estimator still performs significantly better. A.2 Kernel quantile regression For the check loss, we consider the following two examples.

Such a phenomenon is illustrated in Figure 1 where the learned predictive function from the source data may significantly differ from the true function.

However, sampling is not the only source of uncertainty. In practice, distributions change between locations and across time.

Existing works of coping with covariate shift mainly focused on the "

Dealing with distribution shifts is one of the central challenges for modern machine learning. One fundamental situation is the covariate shift, where the input distributions of data change from the training to testing stages while the input-conditional output distribution remains unchanged. In this paper, we initiate the study of a more challenging scenario --- continuous covariate shift --- in which the test data appear sequentially, and their distributions can shift continuously. Our goal is to adaptively train the predictor such that its prediction risk accumulated over time can be minimized. Starting with the importance-weighted learning, we theoretically show the method works effectively if the time-varying density ratios of test and train inputs can be accurately estimated. However, existing density ratio estimation methods would fail due to data scarcity at each time step. To this end, we propose an online density ratio estimation method that can appropriately reuse historical information. Our method is proven to perform well by enjoying a dynamic regret bound, which finally leads to an excess risk guarantee for the predictor.

A significant obstacle in the development of robust machine learning models is \emph{covariate shift}, a form of distribution shift that occurs when the input distributions of the training and test sets differ while the conditional label distributions remain the same. Despite the prevalence of covariate shift in real-world applications, a theoretical understanding in the context of modern machine learning has remained lacking. In this work, we examine the exact high-dimensional asymptotics of random feature regression under covariate shift and present a precise characterization of the limiting test error, bias, and variance in this setting. Our results motivate a natural partial order over covariate shifts that provides a sufficient condition for determining when the shift will harm (or even help) test performance. We find that overparameterized models exhibit enhanced robustness to covariate shift, providing one of the first theoretical explanations for this ubiquitous empirical phenomenon. Additionally, our analysis reveals an exact linear relationship between the in-distribution and out-of-distribution generalization performance, offering an explanation for this surprising recent observation.

Such failure can be caused by distribution shift (also known as dataset shift) between the training and test datasets. For this reason, distribution shift and domain adaptation (a notion comprising techniques for tackling distribution shift) has been a major research topic in machine learning for some time. This paper takes the perspective of Kouw and Loog (2021) and studies the case where feature observations from the test dataset are available for analysis but observations of labels are missing. Under these circumstances, without any assumptions on the nature of the distribution shift between the training and test datasets meaningful prediction of the labels in the test dataset or of their distribution is not feasible. See Kouw and Loog (2021) for a survey of approaches to domain adaptation and their related assumptions. Arguably, covariate shift (also known as population drift) and label shift (also known as prior probability shift or target shift) are the most popular specific distribution shift assumptions, both for their intuiveness as well as their computational manageability. However, exclusive covariate and label shift assumptions have been criticised for being insufficient for common domain adaptation tasks (e.g.