Asymptotically Efficient Off-Policy Evaluation for Tabular Reinforcement Learning
This work addresses the curse of horizon in reinforcement learning for researchers and practitioners, offering an asymptotically efficient solution for finite action spaces, though it is incremental as it builds on existing MIS approaches.
The paper tackles the problem of off-policy evaluation in reinforcement learning by modifying a marginalized importance sampling estimator to asymptotically achieve the Cramer-Rao lower bound of order Ω(H²/n), improving upon prior methods that had an error of O(H³/n).
We consider the problem of off-policy evaluation for reinforcement learning, where the goal is to estimate the expected reward of a target policy $π$ using offline data collected by running a logging policy $μ$. Standard importance-sampling based approaches for this problem suffer from a variance that scales exponentially with time horizon $H$, which motivates a splurge of recent interest in alternatives that break the "Curse of Horizon" (Liu et al. 2018, Xie et al. 2019). In particular, it was shown that a marginalized importance sampling (MIS) approach can be used to achieve an estimation error of order $O(H^3/ n)$ in mean square error (MSE) under an episodic Markov Decision Process model with finite states and potentially infinite actions. The MSE bound however is still a factor of $H$ away from a Cramer-Rao lower bound of order $Ω(H^2/n)$. In this paper, we prove that with a simple modification to the MIS estimator, we can asymptotically attain the Cramer-Rao lower bound, provided that the action space is finite. We also provide a general method for constructing MIS estimators with high-probability error bounds.