# Switching-Geometry Analysis of Deflated Q-Value Iteration

arXiv: https://arxiv.org/abs/2605.10811 · HTML: https://mlsanity.com/paper/2605.10811/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 5 of 5)

- [A First-Order Approach To Accelerated Value Iteration](https://mlsanity.com/paper/1905.09963/) (2019)
- [Switching Theory for Q-Learning](https://mlsanity.com/paper/2604.19569/) (2026)
- [Deflated Dynamics Value Iteration](https://mlsanity.com/paper/2407.10454/) (2024)
- [Rank-One Modified Value Iteration](https://mlsanity.com/paper/2505.01828/) (2025)
- [Beyond the Bellman Fixed Point: Geometry and Fast Policy Identification in Value Iteration](https://mlsanity.com/paper/2604.17457/) (2026)

## Built on by (most cited first, 1 of 1)

- [Heavy-Ball Q-Learning with Residual Weighting Correction](https://mlsanity.com/paper/2606.27112/) (2026)

## Abstract (arXiv)

This paper develops a joint spectral radius (JSR) framework for analyzing rank-one deflated Q-value iteration (Q-VI) in discounted Markov decision process control. Focusing on an all-ones residual correction, we interpret the resulting algorithm through the geometry of switching systems and, to the best of our knowledge, give the first JSR-based convergence analysis of deflated Q-VI for policy optimization problems. Our analysis reveals that the standard Q-VI switching system model has JSR exactly the discount factor γ∈ (0,1), since all admissible subsystems share the all-ones vector as an invariant direction. By passing to the quotient space that removes this direction, we obtain a projected switching system model whose JSR governs the relevant error dynamics and may be strictly smaller than γ. Therefore, the deflated Q-VI admits a potentially sharper convergence-rate characterization than the ambient-space γ-bound. Finally, we prove that the correction is equivalent to a scalar recentering of standard Q-VI. Hence, the projected trajectory, and therefore the greedy-policy sequence, is unchanged relative to standard Q-VI initialized from the same point. The benefit of deflation is not a change in the induced decision-making problem, but a more precise JSR-based description of the convergence geometry after the redundant all-ones component is removed.