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A Theoretical Analysis of State Similarity Between Markov Decision Processes

Zhenyu Tao, Wei Xu, Xiaohu You

TL;DR

<3-5 sentence high-level summary> This work introduces Generalized Bisimulation Metric (GBSM) to measure state similarity across arbitrary pairs of MDPs. It defines GBSM via a Wasserstein-based, Hausdorff-distance construction and proves key metric properties (GBSM symmetry, inter-MDP triangle inequality, and a distance bound on identical spaces) along with a bound linking value-function differences to GBSM. Leveraging these properties, it derives tighter theoretical bounds for policy transfer, state aggregation, and sampling-based estimation across MDPs, accompanied by a closed-form sample complexity. Numerical experiments on synthetic Garnet MDPs and sim-to-real wireless RL tasks validate the theory and demonstrate that GBSM provides more informative, tighter guarantees than traditional BSM in multi-MDP settings.

Abstract

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.

A Theoretical Analysis of State Similarity Between Markov Decision Processes

TL;DR

<3-5 sentence high-level summary> This work introduces Generalized Bisimulation Metric (GBSM) to measure state similarity across arbitrary pairs of MDPs. It defines GBSM via a Wasserstein-based, Hausdorff-distance construction and proves key metric properties (GBSM symmetry, inter-MDP triangle inequality, and a distance bound on identical spaces) along with a bound linking value-function differences to GBSM. Leveraging these properties, it derives tighter theoretical bounds for policy transfer, state aggregation, and sampling-based estimation across MDPs, accompanied by a closed-form sample complexity. Numerical experiments on synthetic Garnet MDPs and sim-to-real wireless RL tasks validate the theory and demonstrate that GBSM provides more informative, tighter guarantees than traditional BSM in multi-MDP settings.

Abstract

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.

Paper Structure

This paper contains 26 sections, 21 theorems, 78 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  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. Composite Applications
  10. Extensions and Algorithm
  11. Extensions to BSM variants
  12. Efficient Algorithm for Calculating GBSM
  13. Numerical Results
  14. Validation on Synthetic MDPs
  15. Policy Transfer Regret Bound
  16. ...and 11 more sections

Key Result

Theorem 1

Let $d_0^{1\text{-}2}$ be a constant zero function and define Then $d_n^{1\text{-}2}$ converges to the fixed point $d^{1\text{-}2}$ with $n\rightarrow\infty$.

Figures (5)

  • Figure 1: Comparison between BSM and GBSM: (a) Analysis of state similarity within a single MDP using BSM. (b) Analysis of state similarity between two MDPs using BSM (requiring a disjoint union construction). (c) Direct analysis of state similarity between two MDPs using GBSM.
  • Figure 2: Policy transfer experiments on random Garnet MDPs.
  • Figure 3: VFA experiments on random Garnet MDPs.
  • Figure 4: SSA experiments on random Garnet MDPs.
  • Figure 5: Sim-to-real experiment results in wireless network testbed.

Theorems & Definitions (43)

  • Definition 1: GBSM
  • Theorem 1: Existence of GBSM
  • proof : Proof Sketch
  • Theorem 2: Optimal value difference bound between MDPs
  • proof : Proof Sketch
  • Theorem 3: GBSM symmetry
  • proof
  • Theorem 4: Inter-MDP triangle inequality of GBSM
  • proof : Proof Sketch
  • Theorem 5: Distance bound on identical spaces
  • ...and 33 more