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Inferring Transition Dynamics from Value Functions

Jacob Adamczyk

TL;DR

The paper tackles inferring environment transition dynamics from pre-trained value functions in reinforcement learning by rearranging the Bellman equation to recover a dynamics model $s' = f(s,a)$ from $Q^\pi$ and $V^\pi$, under some structural assumptions. It provides theoretical guarantees for both continuous and discrete state spaces: in the continuous case, the next-state estimation error scales with the value-function error as $|s' - \hat s'| < \frac{1+\gamma}{\gamma L} \varepsilon$ where $L$ is the reverse-Lipschitz constant, while in the discrete case identifiability holds if $\varepsilon < \delta (2/\gamma + 2)^{-1}$ for a $\delta$-separable $V^\pi$. The work demonstrates a practical proof-of-concept in a tabular grid world, showing that successor states can be recovered from value functions with accuracy governed by these bounds, and discusses how higher discount factors $\gamma$ improve identifiability. Overall, the approach bridges model-free and model-based RL by reusing offline value solutions to infer dynamics, with potential benefits for offline/transfer learning and sample-efficient planning without explicit dynamics models.

Abstract

In reinforcement learning, the value function is typically trained to solve the Bellman equation, which connects the current value to future values. This temporal dependency hints that the value function may contain implicit information about the environment's transition dynamics. By rearranging the Bellman equation, we show that a converged value function encodes a model of the underlying dynamics of the environment. We build on this insight to propose a simple method for inferring dynamics models directly from the value function, potentially mitigating the need for explicit model learning. Furthermore, we explore the challenges of next-state identifiability, discussing conditions under which the inferred dynamics model is well-defined. Our work provides a theoretical foundation for leveraging value functions in dynamics modeling and opens a new avenue for bridging model-free and model-based reinforcement learning.

Inferring Transition Dynamics from Value Functions

TL;DR

The paper tackles inferring environment transition dynamics from pre-trained value functions in reinforcement learning by rearranging the Bellman equation to recover a dynamics model from and , under some structural assumptions. It provides theoretical guarantees for both continuous and discrete state spaces: in the continuous case, the next-state estimation error scales with the value-function error as where is the reverse-Lipschitz constant, while in the discrete case identifiability holds if for a -separable . The work demonstrates a practical proof-of-concept in a tabular grid world, showing that successor states can be recovered from value functions with accuracy governed by these bounds, and discusses how higher discount factors improve identifiability. Overall, the approach bridges model-free and model-based RL by reusing offline value solutions to infer dynamics, with potential benefits for offline/transfer learning and sample-efficient planning without explicit dynamics models.

Abstract

In reinforcement learning, the value function is typically trained to solve the Bellman equation, which connects the current value to future values. This temporal dependency hints that the value function may contain implicit information about the environment's transition dynamics. By rearranging the Bellman equation, we show that a converged value function encodes a model of the underlying dynamics of the environment. We build on this insight to propose a simple method for inferring dynamics models directly from the value function, potentially mitigating the need for explicit model learning. Furthermore, we explore the challenges of next-state identifiability, discussing conditions under which the inferred dynamics model is well-defined. Our work provides a theoretical foundation for leveraging value functions in dynamics modeling and opens a new avenue for bridging model-free and model-based reinforcement learning.
Paper Structure (16 sections, 6 theorems, 21 equations, 2 figures)

This paper contains 16 sections, 6 theorems, 21 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Background
  4. Reinforcement Learning
  5. Preliminaries
  6. Prior Work
  7. Results
  8. Theory for Continuous Spaces
  9. Theory for Discrete Spaces
  10. Experiments
  11. Discussion
  12. Acknowledgements
  13. Appendix: Proofs
  14. Proof of Theorem \ref{['thm:sp-error']}
  15. Proof of Theorem \ref{['thm:tab-sp']}
  16. ...and 1 more sections

Key Result

Lemma 1

Consider an MDP with reverse$(L_r, L_p)$-Lipschitz rewards and dynamics, $L_\pi$-Lipschitz continuous policy. Then, the corresponding value function $Q^\pi(s,a)$ is reverse Lipschitz continuous with constant

Figures (2)

  • Figure 1: Value separability improves model accuracy. Bars denote the standard error in an average over 20 tasks solved to fixed tolerance (in some cases smaller than thickness of bar). The vertical dashed lines indicate the critical value given by the bound in Theorem \ref{['thm:tab-sp']}.
  • Figure 2: Illustration of the errors described in Theorem \ref{['thm:sp-error']}. The dashed lines represent uncertainty bounds on $V^\pi$ and the scanned value $\mathcal{V}$ from the Bellman equation, forming an uncertain region around the star (the true next-state and corresponding value). Note: although we plot the value as a function of state $s$, in practice this should be interpreted as the space of potential successor states, $s'$.

Theorems & Definitions (11)

  • Definition 1: Reverse Lipschitz continuity
  • Lemma 1: rachelson2010locality
  • Definition 2: $\varepsilon$-Accurate Value
  • Proposition 1
  • Theorem 1
  • Definition 3: $\delta$-Separable Value Function
  • Theorem 2: Successor-State Identifiability
  • Theorem
  • proof
  • Theorem : Successor-State Identifiability
  • ...and 1 more