Switching-Geometry Analysis of Deflated Q-Value Iteration

Problem
This paper addresses a gap in the convergence analysis of rank-one deflated Q-value iteration (Q-VI) within the context of discounted Markov decision processes (MDPs). Specifically, it provides a novel joint spectral radius (JSR) framework for analyzing deflated Q-VI, which has not been previously explored in the literature. The authors assert that their work is the first to apply JSR-based techniques to the convergence properties of deflated Q-VI, particularly in policy optimization scenarios. This is a preprint and has not yet undergone peer review.

Method
The core technical contribution of this work is the introduction of a JSR framework to analyze the convergence of deflated Q-VI. The authors focus on an all-ones residual correction and interpret the algorithm through the lens of switching systems. They demonstrate that the JSR of the standard Q-VI switching system model is equal to the discount factor ( \gamma \in (0,1) ), as all admissible subsystems maintain the all-ones vector as an invariant direction. By transitioning to a quotient space that eliminates this invariant direction, the authors derive a projected switching system model. This model’s JSR governs the relevant error dynamics and can be strictly less than ( \gamma ), allowing for a more refined characterization of the convergence rate. The authors also establish that the deflation process is equivalent to a scalar recentering of the standard Q-VI, ensuring that the projected trajectory and the greedy-policy sequence remain unchanged when initialized from the same point.

Results
The paper provides a theoretical framework rather than empirical results, focusing on the mathematical properties of the JSR in relation to deflated Q-VI. The authors demonstrate that the convergence rate of deflated Q-VI can be characterized more precisely than the ambient-space ( \gamma )-bound, although specific numerical results or comparisons against established baselines are not presented. The implications of this sharper characterization suggest that deflated Q-VI can achieve improved convergence properties in practice, although quantifiable effect sizes are not explicitly detailed in the paper.

Limitations
The authors acknowledge that their analysis is limited to the context of discounted MDPs and does not extend to other types of decision processes. Additionally, they do not address potential computational complexities introduced by the JSR framework or the practical implications of implementing the deflation technique in large-scale problems. The lack of empirical validation or experimental results is a notable limitation, as the theoretical findings remain untested in real-world scenarios.

Why it matters
This work has significant implications for the field of reinforcement learning, particularly in enhancing the understanding of convergence properties in Q-learning algorithms. By providing a JSR-based analysis, the authors offer a new perspective on the convergence dynamics of deflated Q-VI, which could lead to more efficient algorithms in policy optimization tasks. The insights gained from this analysis may inform future research on improving convergence rates in various reinforcement learning frameworks, potentially leading to more robust and efficient decision-making systems.

Authors: Donghwan Lee
Source: arXiv:2605.10811
URL: https://arxiv.org/abs/2605.10811v1