New Mathematical Framework for Bellman Residual Minimization in Reinforcement Learning Control
Researchers have published foundational theoretical results for applying Bellman residual minimization (BRM) to policy optimization in control tasks, an area previously underexplored compared to policy evaluation. BRM is an alternative to dynamic programming that directly minimizes a squared error objective and offers more stable convergence when function approximation is used. The work addresses a gap in reinforcement learning theory by extending BRM analysis from evaluation settings to the more complex problem of optimal control.
A preprint posted to arXiv (cs.LG) by Donghwan Lee and collaborators establishes geometry, stationarity, and convergence results for control Bellman residual minimization, targeting policy optimization in Markov decision problems. While BRM has been studied extensively for policy evaluation—estimating the value of a fixed policy—its application to control, where the goal is to find an optimal policy, has received comparatively little theoretical treatment. The authors note that BRM is generally less popular than dynamic programming due to practical inefficiency and challenges in model-free or reinforcement learning settings. However, BRM offers a notable advantage: more stable convergence behavior when value functions are approximated, a common necessity in large or continuous state spaces. The paper aims to fill this theoretical gap by characterizing the objective landscape and proving convergence guarantees for the control setting. The manuscript has undergone four revisions since its initial submission in January 2026, suggesting ongoing refinement of the results.
What's missing
The paper is a preprint and has not yet undergone formal peer review. The study's own scope is limited to theoretical/foundational results; empirical benchmarks comparing BRM-based control to standard dynamic programming or modern deep RL methods are not described in the abstract, leaving practical performance implications unclear. The generality of the convergence guarantees (e.g., assumptions on the MDP structure, function approximation class) is not detailed in the available abstract.
What different sources said
- arXiv cs.LGCenter
Bellman Residual Minimization for Control: Geometry, Stationarity, and Convergence
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