Soft Robust MDPs and Risk-Sensitive MDPs: Equivalence, Policy Gradient, and Sample Complexity
This work addresses decision-making under uncertainty for researchers and practitioners in reinforcement learning, offering theoretical insights and algorithmic improvements, though it is incremental as it builds on existing robust and risk-sensitive MDP frameworks.
The paper tackles the problem of decision-making under uncertainty by introducing a new formulation for risk-sensitive Markov Decision Processes (MDPs) and establishing its equivalence with soft robust MDPs, leading to the derivation of a policy gradient theorem with proven gradient domination and global convergence in tabular settings, and proposing a sample-based offline learning algorithm with analyzed sample complexity.
Robust Markov Decision Processes (MDPs) and risk-sensitive MDPs are both powerful tools for making decisions in the presence of uncertainties. Previous efforts have aimed to establish their connections, revealing equivalences in specific formulations. This paper introduces a new formulation for risk-sensitive MDPs, which assesses risk in a slightly different manner compared to the classical Markov risk measure (Ruszczyński 2010), and establishes its equivalence with a class of soft robust MDP (RMDP) problems, including the standard RMDP as a special case. Leveraging this equivalence, we further derive the policy gradient theorem for both problems, proving gradient domination and global convergence of the exact policy gradient method under the tabular setting with direct parameterization. This forms a sharp contrast to the Markov risk measure, known to be potentially non-gradient-dominant (Huang et al. 2021). We also propose a sample-based offline learning algorithm, namely the robust fitted-Z iteration (RFZI), for a specific soft RMDP problem with a KL-divergence regularization term (or equivalently the risk-sensitive MDP with an entropy risk measure). We showcase its streamlined design and less stringent assumptions due to the equivalence and analyze its sample complexity