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On Reward Transferability in Adversarial Inverse Reinforcement Learning: Insights from Random Matrix Theory

Yangchun Zhang, Wang Zhou, Yirui Zhou

TL;DR

The paper analyzes reward transferability in AIRL under high-dimensional state spaces, deriving a rank-based transferability condition ${\rm rank}(\mathbf{P}-I)=|\mathcal{S}|-1$ and proving via Random Matrix Theory that this rank holds with high probability even when ${\mathbf{P}}$ is unobservable. It shifts the focus from AIRL design to the RL training variance, showing that on-policy learning in the source and off-policy learning in the target mitigate variance-induced instability, and proposes a hybrid PPO-AIRL + SAC framework that substantially improves reward transfer in complex tasks. Key results include rigorous rank- and spectral analyses under uninformative and informative priors, and empirical demonstrations in 2D maze and Ant domains where PPO-AIRL + SAC approaches an oracle baseline. The findings suggest that AIRL’s transferability pitfalls are largely due to training variance of RL algorithms, and that a carefully configured on-/off-policy hybrid can deliver robust, transferable rewards in high-dimensional environments.

Abstract

In the context of inverse reinforcement learning (IRL) with a single expert, adversarial inverse reinforcement learning (AIRL) serves as a foundational approach to providing comprehensive and transferable task descriptions. However, AIRL faces practical performance challenges, primarily stemming from the framework's overly idealized decomposability condition, the unclear proof regarding the potential equilibrium in reward recovery, or questionable robustness in high-dimensional environments. This paper revisits AIRL in \textbf{high-dimensional scenarios where the state space tends to infinity}. Specifically, we first establish a necessary and sufficient condition for reward transferability by examining the rank of the matrix derived from subtracting the identity matrix from the transition matrix. Furthermore, leveraging random matrix theory, we analyze the spectral distribution of this matrix, demonstrating that our rank criterion holds with high probability even when the transition matrices are unobservable. This suggests that the limitations on transfer are not inherent to the AIRL framework itself, but are instead related to the training variance of the reinforcement learning algorithms employed within it. Based on this insight, we propose a hybrid framework that integrates on-policy proximal policy optimization in the source environment with off-policy soft actor-critic in the target environment, leading to significant improvements in reward transfer effectiveness.

On Reward Transferability in Adversarial Inverse Reinforcement Learning: Insights from Random Matrix Theory

TL;DR

The paper analyzes reward transferability in AIRL under high-dimensional state spaces, deriving a rank-based transferability condition and proving via Random Matrix Theory that this rank holds with high probability even when is unobservable. It shifts the focus from AIRL design to the RL training variance, showing that on-policy learning in the source and off-policy learning in the target mitigate variance-induced instability, and proposes a hybrid PPO-AIRL + SAC framework that substantially improves reward transfer in complex tasks. Key results include rigorous rank- and spectral analyses under uninformative and informative priors, and empirical demonstrations in 2D maze and Ant domains where PPO-AIRL + SAC approaches an oracle baseline. The findings suggest that AIRL’s transferability pitfalls are largely due to training variance of RL algorithms, and that a carefully configured on-/off-policy hybrid can deliver robust, transferable rewards in high-dimensional environments.

Abstract

In the context of inverse reinforcement learning (IRL) with a single expert, adversarial inverse reinforcement learning (AIRL) serves as a foundational approach to providing comprehensive and transferable task descriptions. However, AIRL faces practical performance challenges, primarily stemming from the framework's overly idealized decomposability condition, the unclear proof regarding the potential equilibrium in reward recovery, or questionable robustness in high-dimensional environments. This paper revisits AIRL in \textbf{high-dimensional scenarios where the state space tends to infinity}. Specifically, we first establish a necessary and sufficient condition for reward transferability by examining the rank of the matrix derived from subtracting the identity matrix from the transition matrix. Furthermore, leveraging random matrix theory, we analyze the spectral distribution of this matrix, demonstrating that our rank criterion holds with high probability even when the transition matrices are unobservable. This suggests that the limitations on transfer are not inherent to the AIRL framework itself, but are instead related to the training variance of the reinforcement learning algorithms employed within it. Based on this insight, we propose a hybrid framework that integrates on-policy proximal policy optimization in the source environment with off-policy soft actor-critic in the target environment, leading to significant improvements in reward transfer effectiveness.

Paper Structure

This paper contains 17 sections, 11 theorems, 82 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Backgrounds and Notation
  4. Markov Decision Process
  5. Adversarial Inverse Reinforcement Learning
  6. Notation on Random Matrix Theory
  7. Transferability Condition on Environment Dynamics
  8. Uninformative Prior on $\mathfrak{P}$
  9. Informative Prior on $\mathfrak{P}$
  10. Reward Transferability Analysis
  11. Employing an On-policy or Off-policy RL Algorithm in AIRL?
  12. Hybrid Framework PPO-AIRL + SAC
  13. Experiments
  14. Simulation on Location of Eigenvalues
  15. Transfer Effect of PPO-AIRL + SAC
  16. ...and 2 more sections

Key Result

Theorem 1

Let $r_{gt}(s)$ be a ground truth reward, $p$ is a dynamics model, and $\pi_{p}^{\star}$ is the optimal policy under $r_{gt}$ and $p$. Suppose $r'(s)$ is the reward recovered by AIRL that produces an optimal policy $\pi_{p}^{\star}$ in $p$: If ${\rm rank}(\gamma \mathbf{P}-I)=|\mathcal{S}|-1$, when $\gamma$ approaches $1$, i.e., ${\rm rank}(\mathbf{P}-I)=|\mathcal{S}|-1$, then $r'(s)$ is disentan

Figures (6)

  • Figure 1: Local eigenvalues locations for $QQ^{\top}$. (a) $|\mathcal{S}|=900$, (b) $|\mathcal{S}|=2500$.
  • Figure 2: Singular values behavior of $\tilde{B}+|\mathcal{S}|^{-1} ee^{\top}-I$ and $\mathfrak{P}-I$. (a) $|\mathcal{S}|=900$ for $\tilde{B}+|\mathcal{S}|^{-1} ee^{\top}-I$, (b) $|\mathcal{S}|=2500$ for $\tilde{B}+|\mathcal{S}|^{-1} ee^{\top}-I$, (c) $|\mathcal{S}|=900$ for $\mathfrak{P}-I$, (d) $|\mathcal{S}|=2500$ for $\mathfrak{P}-I$.
  • Figure 3: Singular values behavior of $\mathfrak{P}-I$. (a) and (b): The situation of one barrier, (c) and (d): The situation of two barriers.
  • Figure 4: Two cases of reward transfer scenarios. (a) Reward transfer scenarios under changes in the environment structure (Case 1). Group 1: A reward learned in PointMaze-Right is transferred to PointMaze-Left. Group 2: A reward learned in PointMaze-Right is transferred to PointMaze-Multi. Group 3: A reward learned in PointMaze-Double is transferred to PointMaze-Multi. (b) Reward transfer scenarios under changes in the agent dynamics (Case 2). Group 4: A reward learned in Ant is transferred to Ant-Disabled. Group 5: A reward learned in Ant is transferred to Ant-Lengthened.
  • Figure 5: Transfer performance with five seeds in the target environment.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1: Disentangled rewards fu2018learning
  • Theorem 1
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 11 more