A Generalized Bisimulation Metric of State Similarity between Markov Decision Processes: From Theoretical Propositions to Applications

2025-09-23

AI Summary

This work addresses the lack of rigorous metric foundations for policy transfer across multiple Markov decision processes (MDPs). We propose the first Generalized Bisimulation Metric (G-BSM) for cross-MDP state comparison, formally defined over state pairs from distinct MDPs. G-BSM satisfies symmetry, the triangle inequality, and a bounded state-space distance property—yielding significantly tighter theoretical bounds than conventional bisimulation metrics—and admits a closed-form sample complexity bound. Methodologically, we integrate bisimulation theory, state similarity modeling, and sampling-based estimation to construct a provably convergent cross-task metric framework. Empirical evaluation demonstrates that G-BSM achieves both high accuracy and efficient estimation in policy transfer and state aggregation tasks. To our knowledge, it is the first distance metric for multi-task reinforcement learning endowed with strong theoretical guarantees on correctness, convergence, and statistical efficiency.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Imitation Learning & Inverse Reinforcement Learning

Application Category

• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Extending bisimulation metrics to measure state similarity across different Markov decision processes
• Addressing the lack of rigorous mathematical analysis for multi-MDP similarity metrics
• Providing theoretical foundations for policy transfer and state aggregation between MDPs

Innovation

• Generalized bisimulation metric for MDP pairs
• Rigorous proof of symmetry and triangle inequality
• Closed-form sample complexity for estimation