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On Some Geometric Behavior of Value Iteration on the Orthant: Switching System Perspective

Donghwan Lee

TL;DR

The paper investigates value iteration for discounted MDPs through switching-system theory, linking the Bellman updates to switched affine dynamics to reveal new geometric behavior of the iterates. It introduces a switching-system model of Q-value iteration and provides a novel contraction proof, then leverages Lyapunov analysis in a positive-system setting to show that when the initial iterate lies in the shifted orthant ($Q_0\le Q^*$), the iteration remains in that region and contracts in a weighted Euclidean norm ${\|\cdot\|}_M$, with ellipsoidal sublevel sets and a directional contraction along a positive vector $v$. The work demonstrates both a contraction bound in the classical ${\|\cdot\|}_\infty$ sense and a stronger, geometry-aware bound in ${\|\cdot\|}_M$, enriching the analytical toolkit for Q-learning-style schemes. By bridging switching-system theory and dynamic programming, the paper offers a new lens for understanding convergence and paves the way for applying these methods to broader settings and problems.

Abstract

In this paper, the primary goal is to offer additional insights into the value iteration through the lens of switching system models in the control community. These models establish a connection between value iteration and switching system theory and reveal additional geometric behaviors of value iteration in solving discounted Markov decision problems. Specifically, the main contributions of this paper are twofold: 1) We provide a switching system model of value iteration and, based on it, offer a different proof for the contraction property of the value iteration. 2) Furthermore, from the additional insights, new geometric behaviors of value iteration are proven when the initial iterate lies in a special region. We anticipate that the proposed perspectives might have the potential to be a useful tool, applicable in various settings. Therefore, further development of these methods could be a valuable avenue for future research.

On Some Geometric Behavior of Value Iteration on the Orthant: Switching System Perspective

TL;DR

The paper investigates value iteration for discounted MDPs through switching-system theory, linking the Bellman updates to switched affine dynamics to reveal new geometric behavior of the iterates. It introduces a switching-system model of Q-value iteration and provides a novel contraction proof, then leverages Lyapunov analysis in a positive-system setting to show that when the initial iterate lies in the shifted orthant (), the iteration remains in that region and contracts in a weighted Euclidean norm , with ellipsoidal sublevel sets and a directional contraction along a positive vector . The work demonstrates both a contraction bound in the classical sense and a stronger, geometry-aware bound in , enriching the analytical toolkit for Q-learning-style schemes. By bridging switching-system theory and dynamic programming, the paper offers a new lens for understanding convergence and paves the way for applying these methods to broader settings and problems.

Abstract

In this paper, the primary goal is to offer additional insights into the value iteration through the lens of switching system models in the control community. These models establish a connection between value iteration and switching system theory and reveal additional geometric behaviors of value iteration in solving discounted Markov decision problems. Specifically, the main contributions of this paper are twofold: 1) We provide a switching system model of value iteration and, based on it, offer a different proof for the contraction property of the value iteration. 2) Furthermore, from the additional insights, new geometric behaviors of value iteration are proven when the initial iterate lies in a special region. We anticipate that the proposed perspectives might have the potential to be a useful tool, applicable in various settings. Therefore, further development of these methods could be a valuable avenue for future research.
Paper Structure (14 sections, 11 theorems, 38 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 11 theorems, 38 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Markov decision problem
  5. Switching system
  6. Definitions
  7. Q-value iteration (Q-VI)
  8. Switching system model
  9. Finite-time error bound of Q-VI
  10. Convergence of Q-VI on the orthant
  11. Conclusion
  12. Appendix
  13. Proof of Proposition \ref{['prop:Lyapunov-theorem']}
  14. Proof of Proposition \ref{['prop:Lyapunov-theorem2']}

Key Result

Lemma 1

We have $\|Q^*\|_\infty \leq \frac{1}{1-\gamma}$.

Figures (1)

  • Figure 1: Evolution of $Q_k-Q^*$ and geometric properties from \ref{['thm:4']}.

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Lemma 2
  • Proposition 1
  • Lemma 3
  • proof
  • Proposition 2: Upper and lower bounds
  • proof
  • Proposition 3
  • proof
  • ...and 10 more