We consider off-policy evaluation (OPE) of deterministic target policies for reinforcement learning (RL) in environments with continuous action spaces. While it is common to use importance sampling for OPE, it suffers from high variance when the behavior policy deviates significantly from the target policy. In order to address this issue, some recent works on OPE proposed in-sample learning with importance resampling. Yet, these approaches are not applicable to deterministic target policies for continuous action spaces. To address this limitation, we propose to relax the deterministic target policy using a kernel and learn the kernel metrics that minimize the overall mean squared error of the estimated temporal difference update vector of an action value function, where the action value function is used for policy evaluation. We derive the bias and variance of the estimation error due to this relaxation and provide analytic solutions for the optimal kernel metric. In empirical studies using various test domains, we show that the OPE with in-sample learning using the kernel with optimized metric achieves significantly improved accuracy than other baselines.

We consider local kernel metric learning for off-policy evaluation (OPE) of deterministic policies in contextual bandits with continuous action spaces. Our work is motivated by practical scenarios where the target policy needs to be deterministic due to domain requirements, such as prescription of treatment dosage and duration in medicine. Although importance sampling (IS) provides a basic principle for OPE, it is ill-posed for the deterministic target policy with continuous actions. Our main idea is to relax the target policy and pose the problem as kernel-based estimation, where we learn the kernel metric in order to minimize the overall mean squared error (MSE). We present an analytic solution for the optimal metric, based on the analysis of bias and variance. Whereas prior work has been limited to scalar action spaces or kernel bandwidth selection, our work takes a step further being capable of vector action spaces and metric optimization. We show that our estimator is consistent, and significantly reduces the MSE compared to baseline OPE methods through experiments on various domains.