Toward a Unified Lyapunov-Certified ODE Convergence Analysis of Smooth Q-Learning with p-Norms

Abstract

Convergence of Q-learning has been the subject of extensive study for decades. Among the available techniques, the ordinary differential equation (ODE) method is particularly appealing as a general-purpose, off-the-shelf tool for sanity-checking the convergence of a wide range of reinforcement learning algorithms. In this paper, we develop a unified ODE-based convergence framework that applies to standard Q-learning and several soft/smoothed variants, including those built on the log-sum-exponential softmax, Boltzmann softmax, and mellowmax operators. Our analysis uses a smooth p-norm Lyapunov function, leading to concise yet rigorous stability arguments and circumventing the non-smoothness issues inherent to classical infty-norm-based approaches. To the best of our knowledge, the proposed framework is among the first to provide a unified ODE-based treatment that is broadly applicable to smooth Q-learning algorithms while also encompassing standard Q-learning. Moreover, it remains valid even in settings where the associated Bellman operator is not a contraction, as may happen in Boltzmann soft Q-learning.