Study Analyzes How Graph Connectivity Affects Laplacian-Based State Representations in Reinforcement Learning
Researchers have proven upper bounds on approximation error in reinforcement learning when using Laplacian-based state representations, showing how error scales with the algebraic connectivity of the state graph. The work addresses a core challenge in large-scale RL: learning compact state representations in Markov Decision Processes without knowing the full transition graph. The findings provide a principled, end-to-end theoretical framework for understanding when and why spectral state representations succeed or fail.
A new preprint on arXiv presents theoretical guarantees for Laplacian-based state representations in reinforcement learning, a technique used to combat the curse of dimensionality in large Markov Decision Processes. The authors prove an upper bound on the linear value function approximation error under learned spectral features, demonstrating that this error is governed in part by the algebraic connectivity—a measure of how well-connected the state transition graph is. They further bound the error arising from estimating eigenvectors via sample trajectories rather than from the true graph, yielding a complete end-to-end error decomposition across the representation learning pipeline. Notably, the results hold for general, non-uniform policies and do not require symmetry assumptions on the transition kernel, broadening their applicability. The paper also clarifies the formulation of the Laplacian operator in the RL setting, identifying and correcting what the authors describe as common misunderstandings in prior literature. Theoretical findings are validated through numerical simulations on gridworld environments. The work was submitted in March 2026 and has undergone two revisions, with the latest version posted in June 2026.
What's missing
As a preprint, this work has not yet undergone formal peer review. The empirical validation is limited to gridworld environments, leaving open questions about how the bounds behave in continuous or high-dimensional state spaces common in practical RL applications.
What different sources said
- arXiv cs.LGCenter
Impact of Connectivity on Laplacian Representations in Reinforcement Learning
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