Robust $Q$-Aggregation under Covariate Shift with Pairwise Overlap

We study squared-loss aggregation of a fixed dictionary $\mathcal{F}$ of $M$ predictors under covariate shift: a labeled sample of size $n$ comes from a source population with input law $P$, while the target risk is under another input law $Q$. With known density ratios, the difficulty of a comparison depends on the weights where its two predictors disagree. We capture this dependence through an original-pair overlap coefficient ${\Gamma }_{\mathcal{F}}$ and construct a robust $Q$-aggregate from median-of-means (MOM) estimates of risk differences and squared distances. Coordinatewise robust estimates can produce comparison cycles and non-Euclidean distance matrices. Our method repairs the distances by a root-distance ${\ell }_{\infty }$ projection onto the Euclidean distance matrices, then combines the predictors by solving a monotone variational inequality on the simplex. With exact population quantities, this rule is the classical uniform-prior $Q$-aggregate with mixing parameter $1/2$. With noisy pairwise quantities, the variational form also accepts nonzero comparison cycles. The inputs are the density-ratio weights, a failure probability $\delta$, and a bound $B$ on predictions such that, for a source input-response pair $(X,Y)$, the source conditional second moment satisfies $\mathbb{E}[Y^2\mid X]\le B^2$ wherever candidate predictors disagree. Without a supplied overlap bound, the resulting convex mixture achieves a leading-one oracle inequality of order $B^2\max \left\{1,{\Gamma }_{\mathcal{F}}\right\}\log (M/\delta )/n$, including settings in which the global second moment of the density ratio is infinite. Matching lower bounds show the worst-case dependence on overlap, dictionary size, and confidence.