Switching Theory for Q-Learning

Q-learning is a fundamental algorithmic primitive in reinforcement learning. This paper develops a new framework for analyzing constant step-size tabular Q-learning from a switching linear system (SLS) viewpoint. In particular, we derive a stochastic SLS representation of the Q-learning error, and a finite-time error analysis through the joint spectral radius (JSR) of the corresponding SLS model, where the JSR is the exact worst-case exponential rate of the associated SLS. To the best of our knowledge, this is the first convergence rate analysis of standard Q-learning whose leading exponential rate is expressed through the JSR. The resulting rate is tied to the intrinsic worst-case exponential rate of the direct SLS representation and can be sharper than row-sum upper bounds when those bounds are conservative. We further prove that the JSR of Q-learning equals the largest spectral radius among the deterministic-policy modes and give an exact linear programming characterization that can be evaluated to any prescribed accuracy.