Misspecified setting (r < 1/2) for importance-weighted spectral regression under covariate shift

Establish convergence guarantees for the importance-weighted spectral regression algorithm with estimated (possibly unbounded) density ratios under covariate shift in the misspecified setting where the target regression function does not belong to the reproducing kernel Hilbert space H; equivalently, analyze the case in which the source condition parameter r in the relation f_rho = L_{rho_X}^{r} v with v in L^2(X, rho_X) satisfies r < 1/2.

Background

The paper establishes high-probability convergence rates for covariate shift adaptation by combining a new unbounded density ratio estimator with importance-weighted spectral algorithms, under a standard source condition that places the true regression function in the RKHS (i.e., r ≥ 1/2).

While prior work has considered misspecification in related settings (often assuming known or bounded density ratios), this work’s analysis with estimated, potentially unbounded density ratios under covariate shift is conducted only for the well-specified case. Extending these guarantees to the misspecified regime r < 1/2 would broaden applicability to settings where the regression function lies outside the RKHS.