State-space models through the lens of ensemble control

Abstract:State-space models (SSMs) are effective architectures for sequential modeling, but a rigorous theoretical understanding of their training dynamics is still lacking. We formulate continuous-time SSM parameter-path optimization as an ensemble optimal control problem: each input sequence generates a corresponding state trajectory in the ensemble, while the trainable parameter path forms a common control shared by all trajectories. Within this formulation, model evaluation is represented by the forward state equation and backward sensitivity propagation by the corresponding adjoint equation. We show that the Hamiltonian gradient with respect to the control variable represents the negative first-variation density of the reduced ensemble objective. Using this representation, we analyze a Bregman mirror-descent scheme for the continuous-time ensemble objective; in Euclidean geometry this reduces to functional projected gradient descent. State-adjoint and Hamiltonian-gradient stability yield two-sided relative curvature of the reduced objective and relative convexity under sufficient regularization. We further show that in the relatively convex regime the optimal-control problem admits a minimizer, which is unique under relative strong convexity. For SSMs, the explicit stability-generated defect $\kappa_{\mathrm{stab}}^{\mathrm{SSM}}$ gives an $O(1/k)$ objective rate at $\tau=\kappa_{\mathrm{stab}}^{\mathrm{SSM}}$ and geometric convergence to the unique minimizer for $\tau>\kappa_{\mathrm{stab}}^{\mathrm{SSM}}$, under the stated stepsize condition.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2603.13587 [math.OC]

(or arXiv:2603.13587v2 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2603.13587 Focus to learn more arXiv-issued DOI via DataCite

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