Efficient and Provable Algorithms for Covariate Shift

TL;DR

This paper introduces efficient algorithms with theoretical guarantees for covariate shift, enabling accurate estimation of expectations under test distributions using labeled training data and unlabeled test samples.

Contribution

It provides the first rigorous analysis of algorithms for covariate shift with unrestricted functions, establishing a foundational theoretical framework.

Findings

Algorithms with provable sample complexity and computational guarantees

First rigorous analysis for covariate shift with unrestricted functions

Foundation for future theoretical development in covariate shift algorithms

Abstract

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.