Efficient and Provable Algorithms for Covariate Shift
This work addresses a core subroutine in learning under distributional shift, offering foundational theoretical guarantees for researchers and practitioners dealing with covariate shift.
The paper tackles the problem of estimating the average of an unknown bounded function under covariate shift, providing efficient algorithms with provable sample complexity and computational guarantees. It offers the first rigorous analysis for unrestricted functions, establishing a theoretical foundation for covariate shift problems.
Covariate shift, a widely used assumption in tackling {\it distributional shift} (when training and test distributions differ), focuses on scenarios where the distribution of the labels conditioned on the feature vector is the same, but the distribution of features in the training and test data are different. Despite the significance and extensive work on covariate shift, theoretical guarantees for algorithms in this domain remain sparse. In this paper, we distill the essence of the covariate shift problem and focus on estimating the average $\mathbb{E}_{\tilde{\mathbf{x}}\sim p_{\mathrm{test}}}\mathbf{f}(\tilde{\mathbf{x}})$, of any unknown and bounded function $\mathbf{f}$, given labeled training samples $(\mathbf{x}_i, \mathbf{f}(\mathbf{x}_i))$, and unlabeled test samples $\tilde{\mathbf{x}}_i$; this is a core subroutine for several widely studied learning problems. We give several efficient algorithms, with provable sample complexity and computational guarantees. Moreover, we provide the first rigorous analysis of algorithms in this space when $\mathbf{f}$ is unrestricted, laying the groundwork for developing a solid theoretical foundation for covariate shift problems.