Distributional Off-policy Evaluation with Bellman Residual Minimization

We consider the problem of distributional off-policy evaluation which serves as the foundation of many distributional reinforcement learning (DRL) algorithms. In contrast to most existing works (that rely on supremum-extended statistical distances such as supremum-Wasserstein distance), we study the expectation-extended statistical distance for quantifying the distributional Bellman residuals and show that it can upper bound the expected error of estimating the return distribution. Based on this appealing property, by extending the framework of Bellman residual minimization to DRL, we propose a method called Energy Bellman Residual Minimizer (EBRM) to estimate the return distribution. We establish a finite-sample error bound for the EBRM estimator under the realizability assumption. Furthermore, we introduce a variant of our method based on a multi-step bootstrapping procedure to enable multi-step extension. By selecting an appropriate step level, we obtain a better error bound for this variant of EBRM compared to a single-step EBRM, under some non-realizability settings. Finally, we demonstrate the superior performance of our method through simulation studies, comparing with several existing methods.