Inverting the Bellman Equation: From $Q$-Values to World Models

• URL: http://arxiv.org/abs/2606.21173v1
• Date: Fri, 19 Jun 2026 07:26:14 GMT
• Title: Inverting the Bellman Equation: From $Q$-Values to World Models
• Abstract summary: We prove that value-based agents trained on a sufficiently rich set of reward functions implicitly encode a unique and accurate world model.<n>We find that policies trained exclusively on a ttReacher agent's implicit world model are quasi-optimal on out-of-distribution, velocity-based goals despite position-only training.
• Score: 57.827849584133425
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Model-based and model-free reinforcement learning are traditionally viewed as separate paradigms: instead of learning a model of the transition kernel $P$, model-free agents typically estimate value functions tied to a specific policy and reward. In this paper, we challenge this dichotomy by proving that value-based agents trained on a sufficiently rich set of reward functions, e.g. using goal-conditioned RL, implicitly encode a unique and accurate world model. To extract this model in practice, we introduce \textit{$P$-learning}, an inverse analogue to $Q$-learning that samples from an agent's $Q$-values, policies and rewards to decode its internal model of the environment. We then provide sufficient conditions on the type and number of goals for which agents encode the true kernel $P$, covering both stochastic and deterministic MDPs over finite or continuous state spaces. Even when our assumptions are violated, we empirically demonstrate that agents trained on a handful of reward functions encode accurate dynamics in $\texttt{Reacher}$, $\texttt{MountainCar}$ and stochastic variants of $\texttt{FourRooms}$. Surprisingly, we find that policies trained exclusively on a \texttt{Reacher} agent's implicit world model are quasi-optimal on out-of-distribution, velocity-based goals despite position-only training -- suggesting that agents contain hidden generalisation capabilities and providing a new lens into the connection between model-based, model-free, and goal-conditioned RL.

Related papers

• $V_0$: A Generalist Value Model for Any Policy at State Zero [80.7505802128501]
  Policy methods rely on a baseline to measure the relative advantage of an action.<n>This baseline is typically estimated by a Value Model (Critic) often as large as the policy model itself.<n>We propose a Generalist Value Model capable of estimating the expected performance of any model on unseen prompts.
  arXiv  Detail & Related papers  (2026-02-03T14:35:23Z)
• Intention-Conditioned Flow Occupancy Models [80.42634994902858]
  Large-scale pre-training has fundamentally changed how machine learning research is done today.<n>Applying this same framework to reinforcement learning is appealing because it offers compelling avenues for addressing core challenges in RL.<n>Recent advances in generative AI have provided new tools for modeling highly complex distributions.
  arXiv  Detail & Related papers  (2025-06-10T15:27:46Z)
• Partial Identifiability and Misspecification in Inverse Reinforcement Learning [64.13583792391783]
  The aim of Inverse Reinforcement Learning is to infer a reward function $R$ from a policy $pi$. This paper provides a comprehensive analysis of partial identifiability and misspecification in IRL.
  arXiv  Detail & Related papers  (2024-11-24T18:35:46Z)
• Certifiably Robust Model Evaluation in Federated Learning under Meta-Distributional Shifts [8.700087812420687]
  We provide guarantees for the model's performance on a different, unseen network "B"<n>We show how the principled vanilla DKW bound enables certification of the model's true performance on unseen clients within the same (source) network.
  arXiv  Detail & Related papers  (2024-10-26T18:45:15Z)
• Misspecification in Inverse Reinforcement Learning [80.91536434292328]
  The aim of Inverse Reinforcement Learning (IRL) is to infer a reward function $R$ from a policy $pi$. One of the primary motivations behind IRL is to infer human preferences from human behaviour. This means that they are misspecified, which raises the worry that they might lead to unsound inferences if applied to real-world data.
  arXiv  Detail & Related papers  (2022-12-06T18:21:47Z)
• CostNet: An End-to-End Framework for Goal-Directed Reinforcement Learning [9.432068833600884]
  Reinforcement Learning (RL) is a general framework concerned with an agent that seeks to maximize rewards in an environment. There are two approaches, model-based and model-free reinforcement learning, that show concrete results in several disciplines. This paper introduces a novel reinforcement learning algorithm for predicting the distance between two states in a Markov Decision Process.
  arXiv  Detail & Related papers  (2022-10-03T21:16:14Z)
• Generative Temporal Difference Learning for Infinite-Horizon Prediction [101.59882753763888]
  We introduce the $gamma$-model, a predictive model of environment dynamics with an infinite probabilistic horizon. We discuss how its training reflects an inescapable tradeoff between training-time and testing-time compounding errors.
  arXiv  Detail & Related papers  (2020-10-27T17:54:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.