Unbounded Density Ratio Estimation and Its Application to Covariate Shift Adaptation

This paper focuses on the problem of unbounded density ratio estimation -- an understudied yet critical challenge in statistical learning -- and its application to covariate shift adaptation. Much of the existing literature assumes that the density ratio is either uniformly bounded or unbounded but known exactly. These conditions are often violated in practice, creating a gap between theoretical guarantees and real-world applicability. In contrast, this work directly addresses unbounded density ratios and integrates them into importance weighting for effective covariate shift adaptation. We propose a three-step estimation method that leverages unlabeled data from both the source and target distributions: (1) estimating a relative density ratio; (2) applying a truncation operation to control its unboundedness; and (3) transforming the truncated estimate back into the standard density ratio. The estimated density ratio is then employed as importance weights for regression under covariate shift. We establish rigorous, non-asymptotic convergence guarantees for both the proposed density ratio estimator and the resulting regression function estimator, demonstrating optimal or near-optimal convergence rates. Our findings offer new theoretical insights into density ratio estimation and learning under covariate shift, extending classical learning theory to more practical and challenging scenarios.

Spectral Algorithms in Misspecified Regression: Convergence under Covariate Shift

This paper investigates the convergence properties of spectral algorithms -- a class of regularization methods originating from inverse problems -- under covariate shift. In this setting, the marginal distributions of inputs differ between source and target domains, while the conditional distribution of outputs given inputs remains unchanged. To address this distributional mismatch, we incorporate importance weights, defined as the ratio of target to source densities, into the learning framework. This leads to a weighted spectral algorithm within a nonparametric regression setting in a reproducing kernel Hilbert space (RKHS). More importantly, in contrast to prior work that largely focuses on the well-specified setting, we provide a comprehensive theoretical analysis of the more challenging misspecified case, in which the target function does not belong to the RKHS. Under the assumption of uniformly bounded density ratios, we establish minimax-optimal convergence rates when the target function lies within the RKHS. For scenarios involving unbounded importance weights, we introduce a novel truncation technique that attains near-optimal convergence rates under mild regularity conditions, and we further extend these results to the misspecified regime. By addressing the intertwined challenges of covariate shift and model misspecification, this work extends classical kernel learning theory to more practical scenarios, providing a systematic framework for understanding their interaction.