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

Q-value iteration (Q-VI) is usually analyzed through the $\gamma$-contraction of the Bellman operator. This argument proves convergence to $Q^*$, but it gives only a coarse account of when the induced greedy policy becomes optimal. We study discounted Q-VI as a switching system and focus on the practically optimal solution set (POSS), the set of $Q$-functions whose tie-broken greedy policies are optimal. The main result shows that Q-VI reaches the optimal action class in finite time by entering an invariant tube around $\mathcal X_1=Q^*+\operatorname{span}(\mathbf 1)$, which is contained in the POSS. For every $\varepsilon>0$, the distance to $\mathcal X_1$ satisfies an exponential bound with rate $(\bar\rho+\varepsilon)^k$, where $\bar\rho$ is the joint spectral radius of the projected switching family restricted to directions transverse to $\mathcal X_1$. When $\bar\rho<\gamma$, this transverse convergence is faster than the classical contraction rate. The analysis separates fast policy identification from the subsequent convergence to $Q^*$, which may still be governed by the all-ones mode. We also give spectral and graph-theoretic conditions under which the strict inequality $\bar\rho<\gamma$ holds or fails.