Fitted $Q$ Evaluation Without Bellman Completeness via Stationary Weighting

Fitted $Q$-evaluation (FQE) is a standard regression-based tool for off-policy evaluation, but existing stability guarantees often rely on Bellman completeness, a strong closure condition that can fail under function approximation. We study an alternative route: changing the norm used in the regression step. The policy-evaluation Bellman operator is contractive in the $L^2$ norm induced by the target policy's stationary state-action distribution, whereas standard off-policy FQE projects Bellman targets in the behavior-distribution norm. We propose stationary-weighted FQE, which reweights each Bellman regression by the stationary target-to-behavior density ratio. The method preserves FQE's modular supervised-learning form while aligning the fitted projection with that contractive norm. We prove finite-sample linear convergence to the stationary projected Bellman fixed point under misspecification, without requiring Bellman completeness. The bound separates finite-iteration, statistical, approximation, and weight-estimation errors, and shows that ratio-estimation error is attenuated when the inherent Bellman error is small. Controlled experiments show that stationary weighting can stabilize FQE and reduce value error when behavior-norm regression overemphasizes regions rarely visited by the target policy.