Beyond the Bellman Fixed Point: Geometry and Fast Policy Identification in Value Iteration

Dynamic programming is one of the most fundamental methodologies for solving Markov decision problems. Among its many variants, Q-value iteration (Q-VI) is particularly important due to its conceptual simplicity and its classical contraction-based convergence guarantee. Despite the central role of this contraction property, it does not fully reveal the geometric structure of the Q-VI trajectory. In particular, when one is interested not only in the final limit $Q^*$ but also in when the induced greedy policy becomes effectively optimal, the standard contraction argument provides only a coarse characterization. To formalize this notion, we denote by $\mathcal X^*$ the set of $Q$-functions whose corresponding tie-broken greedy policies are optimal, referred to as the practically optimal solution set (POS). In this paper, we revisit discounted Q-VI through the lens of switching system theory and derive new geometric insights into its behavior. In particular, we show that although Q-VI does not reach $Q^*$ in finite time in general, it identifies the optimal action class in finite time. Furthermore, we prove that the distance from the iterate to a particular subset of $\mathcal X^*$ decays exponentially at a rate governed by the joint spectral radius (JSR) of a restricted switching family. This rate can be strictly faster than the standard $γ$ rate when the restricted JSR is strictly smaller than $γ$, while the convergence of the entire $Q$-function to $Q^*$ can still be dominated by the slower $γ$ mode, where $γ$ denotes the discount factor. These results reveal a two-stage geometric behavior of Q-VI: a fast convergence toward $\mathcal X_1$, followed by a slower convergence toward $Q^*$ in general.