Anytime PAC-Bayes for Constrained Density-Ratio Networks under Covariate Shift

A unified framework for learning under covariate shift, in which a constrained density-ratio network approximates the Radon-Nikodym derivative $r^\star = dP/dQ$ and feeds an anytime PAC-Bayes generalization certificate. A change-of-measure identity decomposes the gap between target risk and importance-weighted source risk into a ratio-bias term and a generalization-gap term. Normalization and moment-matching identities are enforced as hard integral constraints through an augmented-Lagrangian scheme, with a second-moment penalty controlling the effective sample size.