🤖 AI Summary
This work addresses the convergence challenge in unsupervised domain adaptation under covariate shift when the target function lies outside the reproducing kernel Hilbert space (i.e., the misspecified setting). By integrating Tikhonov regularization with Nyström subsampling projection, the paper establishes, for the first time, a high-probability excess risk upper bound for Nyström-type domain adaptation methods in this misspecified regime. Leveraging source conditions, effective dimension estimates, and approximation of the Radon–Nikodym derivative, the proposed approach achieves the same convergence rate as in the well-specified setting, requiring only a minimal number of additional samples even when the Radon–Nikodym derivative is unknown.
📝 Abstract
This paper investigates convergence properties of regularized Nyström subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nyström projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.