Unified ODE Analysis of Smooth Q-Learning Algorithms
Donghwan Lee
TL;DR
The paper develops a general ODE-based framework to analyze the asymptotic convergence of Q-learning and its smooth variants, unifying asynchronous updates under a single theory and avoiding restrictive switching-system conditions. By leveraging a weighted $p$-norm Lyapunov function and contraction properties of smooth max operators, it proves global asymptotic stability for a broad class of ODE models that encompass standard Q-learning and smooth variants such as LSE, mellowmax, and Boltzmann softmax. The results show almost-sure convergence to the corresponding fixed points for max, LSE, and mellowmax, while Boltzmann softmax converges via Robbins–Monro arguments with diminishing bias as $oldsymbol{ ho}$ grows. This unified analysis complements prior switching-system approaches and offers simpler, more general proofs applicable to asynchronous Q-learning and its smooth extensions, with implications for convergence guarantees in tabular settings and guidance for selecting smooth operators. Overall, the framework provides a tractable, principled path to assess convergence of a wide family of Q-learning algorithms using ODE methods and weighted-norm Lyapunov functions.
Abstract
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.