arXiv subjects

Ren-Rui Liu

Publications and source records attributed to Ren-Rui Liu.

3 recordsLinked to original sources

arXiv2026

Learning under Target Shift: Optimal Density Ratio Estimation and Importance-Weighted Regression

We study density ratio estimation and importance-weighted regression under target shift with continuous outputs. Under target shift, the conditional distribution of the inputs given the outputs remains invariant across the training and test distributions, while the output marginal distribution may change. Although this problem has been extensively studied for discrete outputs, the continuous setting is substantially less understood: the importance weights are determined by an unknown density ratio function, for which existing estimation methods lack explicit finite-sample convergence rates. We propose a spectral regularization method in a reproducing kernel Hilbert space (RKHS) for estimating the continuous density ratio from labeled training samples and unlabeled test inputs. Under a source condition with regularity parameter $ι>0$, we establish high-probability finite-sample guarantees and show that the estimator achieves the capacity-independent minimax-optimal RKHS-norm rate $O(n_η^{-ι/(2ι+2)})$. We then incorporate the estimated density ratio into importance-weighted regression and characterize the propagation of density-ratio estimation error to the final predictor. When sufficiently many samples are available for density ratio estimation, the resulting regression estimator attains the minimax-optimal rates of standard kernel regression. These results establish a finite-sample theory for continuous density ratio estimation and importance-weighted learning under target shift.

stat.ML↗

arXiv2026

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.

stat.ML↗

arXiv2025

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.

stat.ML↗