Efficient and Provable Algorithms for Covariate Shift

2025-02-21

AI Summary

This paper addresses unbiased functional mean estimation under covariate shift—where training and test data exhibit differing feature distributions but share identical label-conditional distributions. While existing methods are largely restricted to bounded linear or parametric functions, we provide the first rigorous statistical analysis for **arbitrary unknown bounded functions**, establishing a complete theoretical foundation for this problem. We propose a provably correct algorithm that integrates importance weighting, kernel density estimation, and functional approximation. The method achieves optimal sample complexity $O(1/sqrt{n})$ and polynomial-time computational efficiency. Our theoretical guarantees are empirically validated on both synthetic and real-world datasets, demonstrating substantial improvements in estimation accuracy and robustness over prior approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Large-scale security measurements

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 $mathbb{E}_{ ilde{mathbf{x}}sim p_{mathrm{test}}}mathbf{f}( ilde{mathbf{x}})$, of any unknown and bounded function $mathbf{f}$, given labeled training samples $(mathbf{x}_i, mathbf{f}(mathbf{x}_i))$, and unlabeled test samples $ ilde{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.

Problem

• Algorithms for covariate shift lack theoretical guarantees.
• Estimating unknown function averages under distributional shift.
• Providing rigorous analysis for unrestricted functions in covariate shift.

Innovation

• Covariate shift focus
• Efficient algorithms development
• Unrestricted function analysis