Semiparametric Double Reinforcement Learning with Applications to Long-Term Causal Inference (2501.06926v3)

Abstract: Long-term causal effects often must be estimated from short-term data due to limited follow-up in healthcare, economics, and online platforms. Markov Decision Processes (MDPs) provide a natural framework for capturing such long-term dynamics through sequences of states, actions, and rewards. Double Reinforcement Learning (DRL) enables efficient inference on policy values in MDPs, but nonparametric implementations require strong intertemporal overlap assumptions and often exhibit high variance and instability. We propose a semiparametric extension of DRL for efficient inference on linear functionals of the Q-function--such as policy values--in infinite-horizon, time-homogeneous MDPs. By imposing structural restrictions on the Q-function, our approach relaxes the strong overlap conditions required by nonparametric methods and improves statistical efficiency. Under model misspecification, our estimators target the functional of the best-approximating Q-function, with only second-order bias. We provide conditions for valid inference using sieve methods and data-driven model selection. A central challenge in DRL is the estimation of nuisance functions, such as density ratios, which often involve difficult minimax optimization. To address this, we introduce a novel plug-in estimator based on isotonic Bellman calibration, which combines fitted Q-iteration with an isotonic regression adjustment. The estimator is debiased without requiring estimation of additional nuisance functions and reduces high-dimensional overlap assumptions to a one-dimensional condition. Bellman calibration extends isotonic calibration--widely used in prediction and classification--to the MDP setting and may be of independent interest.

• The paper introduces semiparametric restrictions on the Q-function to relax overlap conditions, leading to more robust policy evaluation in DRL.
• It extends the Adaptive Debiased Machine Learning framework with a novel plug-in estimator via isotonic-calibrated fitted Q-iteration.
• The paper derives an efficient influence function that improves estimation precision and reduces variability in long-term causal inference.

Automatic Double Reinforcement Learning in Semiparametric Markov Decision Processes with Applications to Long-Term Causal Inference

The paper explores advancements in the field of reinforcement learning, particularly focusing on the challenges associated with estimating the value of policies from observational and experimental data. Traditional double reinforcement learning (DRL) methods, which have been used to efficiently estimate policy values from different Markov Decision Processes (MDPs), often rely on significant overlap in state distributions, a condition that is frequently unfulfilled in real-world applications.

To address this, the authors propose enhancements to DRL by incorporating semiparametric models to make inferences on linear functionals of the $Q$-function within infinite-horizon, time-invariant MDPs. By relaxing the stringent overlap conditions through semiparametric restrictions, the approach enhances precision and reduces variability in estimates. A prime example explored is long-term value evaluation under the domain adaptation framework, where short trajectory data from new domains must be analyzed for long-term causal inference.

Key Contributions

1. Identification of Constraints on the $Q$-Function: By imposing semiparametric restrictions on the $Q$-function, the authors aim to achieve efficient inference and relax the overlap condition typically required for the DRL. This results in a more precise estimation process in practical applications where overlap is limited or non-existent.
2. Adaptive Debiased Machine Learning (ADML): The paper extends the ADML framework, which is designed to create nonparametrically valid estimators that can adapt to the functional form of the $Q$-function. This adaptability is especially crucial when dealing with potentially misspecified models.
3. New Estimation Techniques: Introducing a novel adaptive debiased plug-in estimator through isotonic-calibrated fitted Q-iteration highlights the paper. This technique aims to bypass computational challenges associated with traditional min-max objective functions used for debiasing.
4. Efficient Influence Function (EIF): The paper provides a detailed exploration of statistical efficiencies such as deriving the efficient influence function for the targeted parameter $\Psi_H$ and discusses how model constraints impact estimation accuracy and variability.

Implications and Future Work

The implications of this work are multifaceted. Practically, the approach enables more robust policy evaluation in settings where experimental or observational data may not fully conform to the standard assumptions required by traditional DRL methods. By improving the reliability of policy valuation in MDPs with constrained overlap, the study opens possibilities for more accurate decision-making in varied settings—ranging from healthcare to digital platforms—where data sparsity or distributional mismatches are common.

Theoretically, this study paves the way for future research focused on combining ADML with DRL in other complex settings, such as non-stationary environments or cases involving multiple adaptive techniques. Further exploration could lead to generalized frameworks that utilize these advanced learning techniques to tackle a broader class of problems in AI, thus fostering more versatile and effective decision-making tools in uncertain or dynamic settings.

Overall, the paper contributes a significant methodological advancement to the field, offering new means of handling some inherent limitations of previous models, which can have widespread implications across various domains utilizing reinforcement learning for policy decision processes.