Lars van der Laan, David Hubbard, Allen Tran et al.
Double Reinforcement Learning (DRL) enables efficient inference for policy values in nonparametric Markov decision processes (MDPs), but existing methods face two major obstacles: (1) they require stringent intertemporal overlap conditions on state trajectories, and (2) they rely on estimating high-dimensional occupancy density ratios. Motivated by problems in long-term causal inference, we extend DRL to a semiparametric setting and develop doubly robust, automatic estimators for general linear functionals of the Q-function in infinite-horizon, time-homogeneous MDPs. By imposing structure on the Q-function, we relax the overlap conditions required by nonparametric methods and obtain efficiency gains. The second obstacle--density-ratio estimation--typically requires computationally expensive and unstable min-max optimization. To address both challenges, we introduce superefficient nonparametric estimators whose limiting variance falls below the generalized Cramer-Rao bound. These estimators treat the Q-function as a one-dimensional summary of the state-action process, reducing high-dimensional overlap requirements to a single-dimensional condition. The procedure is simple to implement: estimate and calibrate the Q-function using fitted Q-iteration, then plug the result into the target functional, thereby avoiding density-ratio estimation altogether.