End-to-End Prediction-to-Decision Certificates for Inverse Design with Vector-Valued Response Surfaces

We study prediction-to-decision certification for inverse design with vector-valued response surfaces. An unknown response map is estimated from data, a target response is prescribed, and a decision is obtained by minimizing a target-loss function. The main question is how statistical prediction error and approximate global optimization error propagate to the true decision quality. We prove an end-to-end certificate showing that a high-probability uniform response bound and a certified global-search tolerance imply a high-probability bound on the true excess risk of the selected decision. Under a growth condition, the same event also yields an explicit distance-to-argmin bound. We provide finite-sample ordinary least squares response certificates, conditional ridge certificates with explicit bias decomposition, certified Lipschitz branch-and-bound, and polynomial sum-of-squares formulations. The formal certificate is demonstrated on a controlled synthetic benchmark. A clay-coloration example illustrates the workflow, while the pilot measurements are treated only as local forward-color checks.