What are the functionals of the reward that can be computed and optimized exactly in Markov Decision Processes? In the finite-horizon, undiscounted setting, Dynamic Programming (DP) can only handle these operations efficiently for certain classes of statistics. We summarize the characterization of these classes for policy evaluation, and give a new answer for the planning problem. Interestingly, we prove that only generalized means can be optimized exactly, even in the more general framework of Distributional Reinforcement Learning (DistRL).DistRL permits, however, to evaluate other functionals approximately. We provide error bounds on the resulting estimators, and discuss the potential of this approach as well as its limitations.These results contribute to advancing the theory of Markov Decision Processes by examining overall characteristics of the return, and particularly risk-conscious strategies.

翻译：在马尔可夫决策过程中，哪些回报泛函能够被精确计算与优化？在有限时段、无折扣设定下，动态规划仅能高效处理特定类别的统计量。我们总结了策略评估中这些类别的特征，并为规划问题提供了新的答案。有趣的是，我们证明即使在更一般的分布强化学习框架中，只有广义均值才能被精确优化。然而，分布强化学习允许对其他泛函进行近似评估。我们给出了由此产生的估计量的误差界限，并讨论了该方法的前景与局限性。这些结果通过考察回报的整体特征，特别是风险意识的策略，推动了马尔可夫决策过程理论的发展。