A Generalized Bisimulation Metric of State Similarity between Markov Decision Processes: From Theoretical Propositions to Applications
This work addresses a theoretical bottleneck in reinforcement learning for multi-MDP scenarios, offering incremental improvements with explicit bounds and sample complexity.
The authors tackled the challenge of generalizing the bisimulation metric to pairs of Markov decision processes, establishing a generalized version with rigorous mathematical properties and deriving tighter theoretical bounds for applications like policy transfer and state aggregation.
The bisimulation metric (BSM) is a powerful tool for computing state similarities within a Markov decision process (MDP), revealing that states closer in BSM have more similar optimal value functions. While BSM has been successfully utilized in reinforcement learning (RL) for tasks like state representation learning and policy exploration, its application to multiple-MDP scenarios, such as policy transfer, remains challenging. Prior work has attempted to generalize BSM to pairs of MDPs, but a lack of rigorous analysis of its mathematical properties has limited further theoretical progress. In this work, we formally establish a generalized bisimulation metric (GBSM) between pairs of MDPs, which is rigorously proven with the three fundamental properties: GBSM symmetry, inter-MDP triangle inequality, and the distance bound on identical state spaces. Leveraging these properties, we theoretically analyse policy transfer, state aggregation, and sampling-based estimation in MDPs, obtaining explicit bounds that are strictly tighter than those derived from the standard BSM. Additionally, GBSM provides a closed-form sample complexity for estimation, improving upon existing asymptotic results based on BSM. Numerical results validate our theoretical findings and demonstrate the effectiveness of GBSM in multi-MDP scenarios.