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arXiv · 2605.11021

A Switching System Theory of Q-Learning with Linear Function Approximation

Donghwan Lee·Han-Dong Lim
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Abstract

Q-learning is a fundamental algorithmic primitive in reinforcement learning. This paper develops a new framework for analyzing linear Q-learning from a switching linear system (SLS) viewpoint, where linear Q-learning denotes Q-learning with linear function approximation. We derive a stochastic SLS representation of the linear Q-learning error and obtain a finite-time error analysis for linear Q-learning through the joint spectral radius (JSR) of the associated SLS family; the JSR is the exact worst-case exponential rate of the corresponding SLSs. The JSR-based rate is tied to the intrinsic worst-case exponential rate of the SLS representation. Moreover, we provide a JSR-based certificate for convergence of linear Q-learning, which can be less conservative than one-step norm bounds.

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Donghwan Lee, Han-Dong Lim. 2026-05-10. A Switching System Theory of Q-Learning with Linear Function Approximation. https://arxiv.org/abs/2605.11021

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