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

Zhenyu Tao, Wei Xu, Xiaohu You

TL;DR

This work defines a generalized bisimulation metric (GBSM) to measure state similarity across pairs of MDPs, and proves three core properties: GBSM symmetry, inter-MDP triangle inequality, and a distance bound on identical state spaces, enabling rigorous multi-MDP analyses. It provides a fixed-point construction for the GBSM distance using the Wasserstein metric and establishes tighter theoretical bounds for policy transfer, state aggregation, and sampling-based estimation than those obtained with standard BSM, including a closed-form sample complexity. Extensions to lax and on-policy GBSM broaden applicability to differing action spaces and non-optimal policies, while numerical results on Garnet MDPs validate the theoretical gains and demonstrate tighter bounds in multi-MDP scenarios. The work highlights practical applications in sim-to-real transfer and multi-task RL, and notes limitations tied to discounting and future work toward average-reward formulations. Overall, GBSM offers a principled, scalable tool for quantifying and leveraging state similarity across multiple MDPs in RL tasks.

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.

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

TL;DR

This work defines a generalized bisimulation metric (GBSM) to measure state similarity across pairs of MDPs, and proves three core properties: GBSM symmetry, inter-MDP triangle inequality, and a distance bound on identical state spaces, enabling rigorous multi-MDP analyses. It provides a fixed-point construction for the GBSM distance using the Wasserstein metric and establishes tighter theoretical bounds for policy transfer, state aggregation, and sampling-based estimation than those obtained with standard BSM, including a closed-form sample complexity. Extensions to lax and on-policy GBSM broaden applicability to differing action spaces and non-optimal policies, while numerical results on Garnet MDPs validate the theoretical gains and demonstrate tighter bounds in multi-MDP scenarios. The work highlights practical applications in sim-to-real transfer and multi-task RL, and notes limitations tied to discounting and future work toward average-reward formulations. Overall, GBSM offers a principled, scalable tool for quantifying and leveraging state similarity across multiple MDPs in RL tasks.

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.

Paper Structure

This paper contains 19 sections, 19 theorems, 76 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Generalized Bisimulation Metric
  4. Applications of GBSM in Multi-MDP Analysis
  5. Performance Bound of Policy Transfer Using GBSM
  6. Approximation Methods and Corresponding Error Bounds
  7. State Aggregation
  8. Sampling-based Estimation
  9. Extensions to BSM variants
  10. Numerical Results
  11. Conslusion
  12. Proof of Theorem \ref{['Theorem:1']}
  13. The Existence of $d^{1\text{-}2}$
  14. The Convergence of $d_n^{1\text{-}2}$ to $d^{1\text{-}2}$
  15. Proof of Theorem \ref{['Theorem:2']}
  16. ...and 4 more sections

Key Result

Theorem 3.2

Let $d_0^{1\text{-}2}$ be a constant zero function and define Then $d_n^{1\text{-}2}$ converges to the fixed point $d^{1\text{-}2}$ uniformly with $n\rightarrow\infty$. Let $\bar{R}=\max_{s,s',a} |R_1(s,a)-R_2(s',a)|$, and the convergence of $d_n^{1\text{-}2}$ to $d^{1\text{-}2}$ satisfies

Figures (13)

  • Figure 1: Experiments on random Garnet MDPs.
  • Figure 2: Experiments on random Garnet MDPs (policy transfer, $\gamma=0.1$ to $0.3$).
  • Figure 3: Experiments on random Garnet MDPs (policy transfer, $\gamma=0.4$ to $0.6$).
  • Figure 4: Experiments on random Garnet MDPs (policy transfer, $\gamma=0.7$ to $0.9$).
  • Figure 5: Experiments on random Garnet MDPs (SSA with aggregation, $\gamma=0.1$ to $0.3$).
  • ...and 8 more figures

Theorems & Definitions (38)

  • Definition 3.1: Generalized bisimulation metric
  • Theorem 3.2: Existence and convergence of GBSM
  • proof : Proof Sketch
  • Theorem 3.3: Optimal value difference bound between MDPs
  • proof : Proof Sketch
  • Theorem 3.4: GBSM symmetry
  • proof
  • Theorem 3.5: Inter-MDP triangle inequality of GBSM
  • proof
  • Theorem 3.6: Distance bound on identical state spaces
  • ...and 28 more