Theoretical guarantees in reinforcement learning (RL) are known to suffer multiplicative blow-up factors with respect to the misspecification error of function approximation. Yet, the nature of such \emph{approximation factors} -- especially their optimal form in a given learning problem -- is poorly understood. In this paper we study this question in linear off-policy value function estimation, where many open questions remain. We study the approximation factor in a broad spectrum of settings, such as with the weighted $L_2$-norm (where the weighting is the offline state distribution), the $L_\infty$ norm, the presence vs. absence of state aliasing, and full vs. partial coverage of the state space. We establish the optimal asymptotic approximation factors (up to constants) for all of these settings. In particular, our bounds identify two instance-dependent factors for the $L_2(\mu)$ norm and only one for the $L_\infty$ norm, which are shown to dictate the hardness of off-policy evaluation under misspecification.

翻译：在强化学习（RL）中，理论保证已知会因函数近似的误指定误差而承受乘法放大因子。然而，此类近似因子的本质——尤其是在给定学习问题中的最优形式——尚不明确。本文在线性离策略值函数估计的背景下研究该问题，其中仍存在许多未解难题。我们在一系列广泛设置中研究近似因子，包括加权$L_2$范数（其中权重为离线状态分布）、$L_\infty$范数、状态别名的有无，以及状态空间的完全覆盖与部分覆盖。我们为所有这些设置建立了最优渐近近似因子（精确至常数）。特别地，我们的界限识别出$L_2(\mu)$范数存在两个实例相关因子，而$L_\infty$范数仅有一个，这些因子被证明决定了误指定条件下离策略评估的难度。