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A Note on Loss Functions and Error Compounding in Model-based Reinforcement Learning

Nan Jiang

TL;DR

This note analyzes why model-based RL often exhibits poor empirical error propagation despite strong theoretical guarantees from planning and OPE. It grounds the discussion in the Simulation Lemma, which shows linear-in-horizon error accumulation under bounded one-step model error, and explains why popular losses like L2 on raw observations or TV-based metrics can misalign with these guarantees. It provides concrete counterexamples for the MuZero (multi-step reward prediction) loss, showing failures in stochastic environments and poor finite-sample behavior under limited coverage, and highlights the pitfalls of latent-learning losses such as bisimulation when latent dynamics are stochastic. The paper advocates focusing on the properties of the learned model $M$ (e.g., smoothness of $V_M^\pi$, quality of latent representations, and appropriate coverage) and suggests that regularization and problem-structure assumptions are essential for meaningful guarantees in deep model-based RL.

Abstract

This note clarifies some confusions (and perhaps throws out more) around model-based reinforcement learning and their theoretical understanding in the context of deep RL. Main topics of discussion are (1) how to reconcile model-based RL's bad empirical reputation on error compounding with its superior theoretical properties, and (2) the limitations of empirically popular losses. For the latter, concrete counterexamples for the "MuZero loss" are constructed to show that it not only fails in stochastic environments, but also suffers exponential sample complexity in deterministic environments when data provides sufficient coverage.

A Note on Loss Functions and Error Compounding in Model-based Reinforcement Learning

TL;DR

This note analyzes why model-based RL often exhibits poor empirical error propagation despite strong theoretical guarantees from planning and OPE. It grounds the discussion in the Simulation Lemma, which shows linear-in-horizon error accumulation under bounded one-step model error, and explains why popular losses like L2 on raw observations or TV-based metrics can misalign with these guarantees. It provides concrete counterexamples for the MuZero (multi-step reward prediction) loss, showing failures in stochastic environments and poor finite-sample behavior under limited coverage, and highlights the pitfalls of latent-learning losses such as bisimulation when latent dynamics are stochastic. The paper advocates focusing on the properties of the learned model (e.g., smoothness of , quality of latent representations, and appropriate coverage) and suggests that regularization and problem-structure assumptions are essential for meaningful guarantees in deep model-based RL.

Abstract

This note clarifies some confusions (and perhaps throws out more) around model-based reinforcement learning and their theoretical understanding in the context of deep RL. Main topics of discussion are (1) how to reconcile model-based RL's bad empirical reputation on error compounding with its superior theoretical properties, and (2) the limitations of empirically popular losses. For the latter, concrete counterexamples for the "MuZero loss" are constructed to show that it not only fails in stochastic environments, but also suffers exponential sample complexity in deterministic environments when data provides sufficient coverage.
Paper Structure (16 sections, 3 theorems, 9 equations, 1 figure)

This paper contains 16 sections, 3 theorems, 9 equations, 1 figure.

Table of Contents

  1. Background
  2. Setup and Notation
  3. Markov Decision Processes (MDPs)
  4. Learning Settings
  5. Planning and Off-Policy Evaluation
  6. Error Compounding?
  7. Deterministic models and $L_2$ loss
  8. Smoothness
  9. Linear control problems
  10. Latent Models and Non-observation-prediction Losses
  11. Bisimulation loss
  12. Multi-step reward prediction loss
  13. True model does not minimize the loss in stochastic environments
  14. Exponential distribution shift
  15. Other losses
  16. ...and 1 more sections

Key Result

Lemma 1

For any $P: \mathcal{S}\times \mathcal{A}\to \Delta(\mathcal{S})$ and any $\pi: \mathcal{S}\to \Delta(\mathcal{A})$, let $J_M(\pi)$ be the expected return of $\pi$ in $M$ specified by $(P, R^\star)$. Then, where $d_{M^\star}^\pi$ is the normalized discounted state-action occupancy induced from $\pi$ in $M^\star$.

Figures (1)

  • Figure 1: Construction for Proposition \ref{['prop:muzero_stoch']}. The top state is the initial state, where both actions lead to the same distribution over A, B, C. The numbers at the second level indicate rewards for taking left (L) and right (R) actions from states A, B, and C, respectively.

Theorems & Definitions (5)

  • Lemma 1: Simulation Lemma
  • Proposition 2
  • proof
  • Proposition 3
  • proof