Unified ODE Analysis of Smooth Q-Learning Algorithms
This work addresses a theoretical bottleneck for researchers in reinforcement learning by providing a simpler and more general convergence proof framework, though it is incremental as it builds on prior ODE and Lyapunov-based methods.
The paper tackles the problem of proving convergence for Q-learning and its smooth variants by introducing a more general and unified ODE-based analysis that improves upon restrictive switching system approaches, enabling broader applicability without needing conditions like quasi-monotonicity.
Convergence of Q-learning has been the focus of extensive research over the past several decades. Recently, an asymptotic convergence analysis for Q-learning was introduced using a switching system framework. This approach applies the so-called ordinary differential equation (ODE) approach to prove the convergence of the asynchronous Q-learning modeled as a continuous-time switching system, where notions from switching system theory are used to prove its asymptotic stability without using explicit Lyapunov arguments. However, to prove stability, restrictive conditions, such as quasi-monotonicity, must be satisfied for the underlying switching systems, which makes it hard to easily generalize the analysis method to other reinforcement learning algorithms, such as the smooth Q-learning variants. In this paper, we present a more general and unified convergence analysis that improves upon the switching system approach and can analyze Q-learning and its smooth variants. The proposed analysis is motivated by previous work on the convergence of synchronous Q-learning based on $p$-norm serving as a Lyapunov function. However, the proposed analysis addresses more general ODE models that can cover both asynchronous Q-learning and its smooth versions with simpler frameworks.