Final Iteration Convergence Bound of Q-Learning: Switching System Approach
Donghwna Lee
TL;DR
This paper casts Q-learning within a discrete-time switching-system framework to derive finite-time bounds for the final iterate. By formulating Q-learning as a stochastic affine switching system and introducing lower and upper comparison systems, the authors obtain a tractable finite-time bound on the final iterate error that decays with a rate $\\rho=1-\\alpha d_{min}(1-\\gamma)$ and depends on problem dimensions, the step-size, and occupation frequencies. The analysis delivers a concrete $\\|\\cdot\\|_\\infty$ bound with a constant bias plus two decaying terms, and provides a complementary bound on the gap between the comparison systems, enabling a full convergence guarantee for the original system under i.i.d. observations and constant step-size. The work highlights the value of control-theoretic switching-system insights for RL finite-time analysis, offering a potentially general template for analyzing more complex RL algorithms, while noting limitations for time-varying behavior policies and non-i.i.d. data scenarios.
Abstract
Q-learning is known as one of the fundamental reinforcement learning (RL) algorithms. Its convergence has been the focus of extensive research over the past several decades. Recently, a new finitetime error bound and analysis for Q-learning was introduced using a switching system framework. This approach views the dynamics of Q-learning as a discrete-time stochastic switching system. The prior study established a finite-time error bound on the averaged iterates using Lyapunov functions, offering further insights into Q-learning. While valuable, the analysis focuses on error bounds of the averaged iterate, which comes with the inherent disadvantages: it necessitates extra averaging steps, which can decelerate the convergence rate. Moreover, the final iterate, being the original format of Q-learning, is more commonly used and is often regarded as a more intuitive and natural form in the majority of iterative algorithms. In this paper, we present a finite-time error bound on the final iterate of Q-learning based on the switching system framework. The proposed error bounds have different features compared to the previous works, and cover different scenarios. Finally, we expect that the proposed results provide additional insights on Q-learning via connections with discrete-time switching systems, and can potentially present a new template for finite-time analysis of more general RL algorithms.