Input-to-State Stability of Gradient Flows in Distributional Space

Abstract: This paper proposes a new notion of distributional Input-to-State Stability (dISS) for dynamic systems evolving in probability spaces over a domain. Unlike other norm-based ISS concepts, we rely on the Wasserstein metric, which captures more precisely the effects of the disturbances on atomic and non-atomic measures. We show how dISS unifies both ISS and Noise to State Stability (NSS) over compact domains for particle dynamics, while extending the classical notions to sets of probability distributions. We then apply the dISS framework to study the robustness of various Wasserstein gradient flows with respect to perturbations. In particular, we establish dISS for gradient flows defined by a class of $l$-smooth functionals subject to bounded disturbances, such as those induced by entropy in optimal transport. Further, we study the dISS robustness of the large-scale algorithms when using Kernel and sample-based approximations. This results into a characterization of the error incurred when using a finite number of agents, which can guide the selection of the swarm size to achieve a mean-field objective with prescribed accuracy and stability guarantees.