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Fuz-RL: A Fuzzy-Guided Robust Framework for Safe Reinforcement Learning under Uncertainty

Xu Wan, Chao Yang, Cheng Yang, Jie Song, Mingyang Sun

TL;DR

Fuz-RL is proposed, a fuzzy measure-guided robust framework for safe RL that develops a novel fuzzy Bellman operator for estimating robust value functions using Choquet integrals and proves that solving the Fuz-RL problem is equivalent to solving distributionally robust safe RL problems (in robust CMDP form), effectively avoiding min-max optimization.

Abstract

Safe Reinforcement Learning (RL) is crucial for achieving high performance while ensuring safety in real-world applications. However, the complex interplay of multiple uncertainty sources in real environments poses significant challenges for interpretable risk assessment and robust decision-making. To address these challenges, we propose Fuz-RL, a fuzzy measure-guided robust framework for safe RL. Specifically, our framework develops a novel fuzzy Bellman operator for estimating robust value functions using Choquet integrals. Theoretically, we prove that solving the Fuz-RL problem (in Constrained Markov Decision Process (CMDP) form) is equivalent to solving distributionally robust safe RL problems (in robust CMDP form), effectively avoiding min-max optimization. Empirical analyses on safe-control-gym and safety-gymnasium scenarios demonstrate that Fuz-RL effectively integrates with existing safe RL baselines in a model-free manner, significantly improving both safety and control performance under various types of uncertainties in observation, action, and dynamics.

Fuz-RL: A Fuzzy-Guided Robust Framework for Safe Reinforcement Learning under Uncertainty

TL;DR

Fuz-RL is proposed, a fuzzy measure-guided robust framework for safe RL that develops a novel fuzzy Bellman operator for estimating robust value functions using Choquet integrals and proves that solving the Fuz-RL problem is equivalent to solving distributionally robust safe RL problems (in robust CMDP form), effectively avoiding min-max optimization.

Abstract

Safe Reinforcement Learning (RL) is crucial for achieving high performance while ensuring safety in real-world applications. However, the complex interplay of multiple uncertainty sources in real environments poses significant challenges for interpretable risk assessment and robust decision-making. To address these challenges, we propose Fuz-RL, a fuzzy measure-guided robust framework for safe RL. Specifically, our framework develops a novel fuzzy Bellman operator for estimating robust value functions using Choquet integrals. Theoretically, we prove that solving the Fuz-RL problem (in Constrained Markov Decision Process (CMDP) form) is equivalent to solving distributionally robust safe RL problems (in robust CMDP form), effectively avoiding min-max optimization. Empirical analyses on safe-control-gym and safety-gymnasium scenarios demonstrate that Fuz-RL effectively integrates with existing safe RL baselines in a model-free manner, significantly improving both safety and control performance under various types of uncertainties in observation, action, and dynamics.
Paper Structure (39 sections, 8 theorems, 46 equations, 14 figures, 4 tables, 1 algorithm)

This paper contains 39 sections, 8 theorems, 46 equations, 14 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Robust Approaches in Safe RL.
  4. Fuzzy Measures in MDPs.
  5. Preliminary
  6. Robust CMDP
  7. Fuzzy Measures Fundamentals
  8. Fuzzy Measure-based Robust Safe RL Framework
  9. Theoretical Foundation of Fuz-RL
  10. Fuzzy Bellman Operator.
  11. Robust Equivalence.
  12. Practical Implementation of Fuz-RL
  13. Estimation of Fuzzy Measures.
  14. Estimation of Choquet Integrals.
  15. Value Network Updates.
  16. ...and 24 more sections

Key Result

Lemma 3.3

For any bounded measurable function $f: \Omega \rightarrow \mathbb{R}$ and $\lambda$-fuzzy measure $m$ with $\lambda \geq 0$: where $\text{core}(m) = \{ P \in \mathcal{P}(\Omega): P(A) \geq m(A) \}$ is the set of probability measures dominating $m$.

Figures (14)

  • Figure 1: Training Dynamics of PPOLag and Fuz-PPOLag under multi-source uncertainty on Safety-Gymnasium tasks. The perturbation intensity during training is set to $\varepsilon = 0.5$.
  • Figure 2: Test Comparison of PPOLag and Fuz-PPOLag under multi-source uncertainty setting over 5 Episodes and 5 seeds on Safety-Gymnasium tasks. The cost_limit is set to 0.1.
  • Figure 3: Ablation study of the uncertainty level $K$.
  • Figure 4: Schematics, state and input vectors of the cart-pole, and the 1D and 2D quadrotor environments in safe-control-gym.
  • Figure 5: (a) Hierarchical relationship of safety sets, (b) Cost space and (c) Reward space trajectory comparisons, with dashed lines indicating safety boundaries.
  • ...and 9 more figures

Theorems & Definitions (15)

  • Definition 3.1: Fuzzy Measure murofushi2000fuzzy
  • Definition 3.2: $\lambda$-Fuzzy Measure denneberg1994non
  • Lemma 3.3: Choquet Integral Representation gilboa1994additive
  • Definition 4.1: Fuzzy Bellman Operator
  • Theorem 4.2: $\gamma$-contraction of Fuzzy Bellman Operator
  • Theorem 4.3: Convergence of Fuzzy Bellman Operator
  • Theorem 4.4: Equivalent Theorem
  • Theorem A.1: $\gamma$-contraction of Fuzzy Bellman Operator
  • proof
  • Theorem A.2: Convergence of Fuzzy Bellman Operator
  • ...and 5 more