Optimizing static risk-averse objectives in Markov decision processes is difficult because they do not admit standard dynamic programming equations common in Reinforcement Learning (RL) algorithms. Dynamic programming decompositions that augment the state space with discrete risk levels have recently gained popularity in the RL community. Prior work has shown that these decompositions are optimal when the risk level is discretized sufficiently. However, we show that these popular decompositions for Conditional-Value-at-Risk (CVaR) and Entropic-Value-at-Risk (EVaR) are inherently suboptimal regardless of the discretization level. In particular, we show that a saddle point property assumed to hold in prior literature may be violated. However, a decomposition does hold for Value-at-Risk and our proof demonstrates how this risk measure differs from CVaR and EVaR. Our findings are significant because risk-averse algorithms are used in high-stake environments, making their correctness much more critical.

翻译：优化马尔可夫决策过程中的静态风险厌恶目标具有挑战性，因为这些目标不满足强化学习算法中常见的标准动态规划方程。近年来，通过离散风险水平扩充状态空间的动态规划分解方法在强化学习领域逐渐流行。先前研究表明，当风险水平充分离散化时，这些分解方法是最优的。然而，我们证明条件风险价值（CVaR）和熵风险价值（EVaR）的流行分解方法本质上存在次优性——无论离散化程度如何。特别地，我们发现前人文献假设的鞍点性质可能被违反。但风险价值（VaR）确实存在有效分解，我们的证明阐明了该风险测度与CVaR及EVaR的根本差异。本研究的重大意义在于：风险厌恶算法被应用于高风险环境，其正确性至关重要。