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A Discrete-Time Switching System Analysis of Q-learning

Donghwan Lee, Jianghai Hu, Niao He

TL;DR

This work reframes asynchronous Q-learning with a constant step-size as a discrete-time stochastic affine switching system and analyzes it via two comparison systems: a lower linear system and an upper linear switching system. By bounding the original dynamics with these tractable systems and applying Lyapunov stability analysis, it derives finite-time error bounds and clarifies the origin of maximization bias in Q-learning. The results yield explicit finite-time guarantees under standard assumptions and illuminate a control-theoretic path for understanding and extending Q-learning variants. Overall, the discrete-time switching-system perspective provides a complementary, intuitive foundation that could enable refined analyses and new algorithm designs in reinforcement learning.

Abstract

This paper develops a novel control-theoretic framework to analyze the non-asymptotic convergence of Q-learning. We show that the dynamics of asynchronous Q-learning with a constant step-size can be naturally formulated as a discrete-time stochastic affine switching system. Moreover, the evolution of the Q-learning estimation error is over- and underestimated by trajectories of two simpler dynamical systems. Based on these two systems, we derive a new finite-time error bound of asynchronous Q-learning when a constant stepsize is used. Our analysis also sheds light on the overestimation phenomenon of Q-learning. We further illustrate and validate the analysis through numerical simulations.

A Discrete-Time Switching System Analysis of Q-learning

TL;DR

This work reframes asynchronous Q-learning with a constant step-size as a discrete-time stochastic affine switching system and analyzes it via two comparison systems: a lower linear system and an upper linear switching system. By bounding the original dynamics with these tractable systems and applying Lyapunov stability analysis, it derives finite-time error bounds and clarifies the origin of maximization bias in Q-learning. The results yield explicit finite-time guarantees under standard assumptions and illuminate a control-theoretic path for understanding and extending Q-learning variants. Overall, the discrete-time switching-system perspective provides a complementary, intuitive foundation that could enable refined analyses and new algorithm designs in reinforcement learning.

Abstract

This paper develops a novel control-theoretic framework to analyze the non-asymptotic convergence of Q-learning. We show that the dynamics of asynchronous Q-learning with a constant step-size can be naturally formulated as a discrete-time stochastic affine switching system. Moreover, the evolution of the Q-learning estimation error is over- and underestimated by trajectories of two simpler dynamical systems. Based on these two systems, we derive a new finite-time error bound of asynchronous Q-learning when a constant stepsize is used. Our analysis also sheds light on the overestimation phenomenon of Q-learning. We further illustrate and validate the analysis through numerical simulations.

Paper Structure

This paper contains 18 sections, 9 theorems, 93 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov decision problem
  4. Switching system
  5. Revisit Q-learning
  6. Assumptions and definitions
  7. Finite-time Analysis of Q-learning from Switching System Theory
  8. Q-learning as a stochastic affine switching system
  9. Lower comparison system
  10. Upper comparison system
  11. Finite-time error bound of Q-learning
  12. Remarks
  13. Sample complexity
  14. Example
  15. Conclusion
  16. ...and 3 more sections

Key Result

Lemma 1

If the step-size is less than one, then for all $k \ge 0$, From assumption:bounded-reward and assumption:bounded-Q0, we can easily see that $Q_{\max}\leq\frac{1}{1-\gamma}$.

Figures (5)

  • Figure 1: Overview of the proposed analysis
  • Figure 2: Evolution of $Q_k-Q^*$ (black solid lines), lower comparison system $Q_k^L-Q^*$ (blue solid lines), and upper comparison system $Q_k^U-Q^*$ (red solid lines) with step-size $\alpha = 0.002$.
  • Figure 3: Evolution of error $Q_k^U-Q_k^L$ (black solid lines) with step-size $\alpha = 0.002$.
  • Figure 4: Evolution of $Q_k-Q^*$ (black solid lines), lower comparison system $Q_k^L-Q^*$ (blue solid lines), and upper comparison system $Q_k^U-Q^*$ (red solid lines) with step-size $\alpha = 0.9$.
  • Figure 5: Evolution of error $Q_k^U-Q_k^L$ (black solid lines) with step-size $\alpha = 0.9$.

Theorems & Definitions (25)

  • Remark 1
  • Definition 1
  • Lemma 1: Boundedness of Q-learning iterates gosavi2006boundedness
  • Remark 2
  • Proposition 1
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Remark 3
  • ...and 15 more