A Theoretical Analysis of State Similarity Between Markov Decision Processes
This work addresses a theoretical gap in reinforcement learning for analyzing state similarities across different MDPs, offering incremental improvements with explicit bounds and sample complexity.
The authors tackled the challenge of measuring state similarity between multiple Markov decision processes (MDPs) by formally establishing a generalized bisimulation metric (GBSM), which provides tighter theoretical bounds for tasks like policy transfer and state aggregation compared to existing methods.
The bisimulation metric (BSM) is a powerful tool for analyzing 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 state similarity between multiple MDPs remains challenging. Prior work has attempted to extend BSM to pairs of MDPs, but a lack of well-established mathematical properties has limited further theoretical analysis between MDPs. In this work, we formally establish a generalized bisimulation metric (GBSM) for measuring state similarity between arbitrary pairs of MDPs, which is rigorously proven with three fundamental metric properties, i.e., GBSM symmetry, inter-MDP triangle inequality, and a distance bound on identical spaces. Leveraging these properties, we theoretically analyze policy transfer, state aggregation, and sampling-based estimation across MDPs, obtaining explicit bounds that are strictly tighter than existing ones 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.