Model Selection for Off-Policy Evaluation via Pairwise Bellman Validation

Lars van der Laan, Vasilis Syrgkanis, Nathan Kallus

Model selection and hyperparameter tuning pose a central challenge in off-policy evaluation (OPE) because errors of fitted Q-functions and occupancy-ratio estimates cannot be directly evaluated from held-out transitions. Bellman-residual and minimax validation methods often require auxiliary regressions or optimization over separate critic classes, adding computational and tuning costs. We introduce pairwise Bellman validation (PBV), a tuning-free minimax procedure for selecting among finite classes of fitted Q-functions and occupancy-ratio estimators using held-out transitions. PBV constructs critics from pairwise candidate differences, requiring only finitely many empirical moment calculations without auxiliary validation models or separate critic optimization. We establish finite-sample oracle inequalities relative to the best available candidate, even under candidate-class misspecification. Our key insight is that Q-function errors and log ratios between candidate and true occupancy ratios provide oracle critics whose Bellman and adjoint Bellman moments control estimation error. Pairwise candidate differences approximate these unknown critics, while penalties account for…