Estimating Unbounded Density Ratios: Applications in Error Control under Covariate Shift
This work addresses error control under covariate shift for statistical and machine learning applications, offering theoretical guarantees for unbounded density ratios, though it is incremental as it builds on existing estimation frameworks.
The paper tackles the problem of estimating density ratios with unbounded domains and ranges, establishing minimax optimal error bounds and applying this to error control in nonparametric regression and conditional flow models under covariate shift, with simulations showing the source estimator outperforming loss correction methods.
The density ratio is an important metric for evaluating the relative likelihood of two probability distributions, with extensive applications in statistics and machine learning. However, existing estimation theories for density ratios often depend on stringent regularity conditions, mainly focusing on density ratio functions with bounded domains and ranges. In this paper, we study density ratio estimators using loss functions based on least squares and logistic regression. We establish upper bounds on estimation errors with standard minimax optimal rates, up to logarithmic factors. Our results accommodate density ratio functions with unbounded domains and ranges. We apply our results to nonparametric regression and conditional flow models under covariate shift and identify the tail properties of the density ratio as crucial for error control across domains affected by covariate shift. We provide sufficient conditions under which loss correction is unnecessary and demonstrate effective generalization capabilities of a source estimator to any suitable target domain. Our simulation experiments support these theoretical findings, indicating that the source estimator can outperform those derived from loss correction methods, even when the true density ratio is known.