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

Guillem Pascual, Sonia Martínez

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.

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 -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.

Paper Structure

This paper contains 18 sections, 11 theorems, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem formulation
  4. dISS for Continuity Equations
  5. dISS notions
  6. Relationship of dISS with microscopic ISS and NSS
  7. Perturbed Gradient Flows
  8. Entropic Disturbances in Optimal Transport
  9. Sample Approximations
  10. Approximations given by KDEs
  11. Semi-discrete Optimal Transport
  12. Numerical Examples
  13. KDE Sinkhorn algorithm
  14. Conclusions
  15. Proof of Lemma \ref{['le:convex']}
  16. ...and 3 more sections

Key Result

Lemma B.8

If the functional $F$ is either $\lambda$ geodesically convex or $\lambda$-displacement convex, it satisfies both quadratic growth and gradient dominance.$\bullet$

Figures (4)

  • Figure 3: The swarm relaxes from an initial skewed, high-concentration state $\rho_{t_1}$ (left), through a transitional state $\rho_{t_2}$ (middle), moving towards the target distribution $\rho^*$ (right). In this case $W_2(\rho_{t_2},\rho^*)< W_2(\rho_{t_1},\rho^*)$; however $\|\rho_{t_1}-\rho^*\|_{L^2} = \|\rho_{t_2}-\rho^*\|_{L^2}$, due to non-overlapping supports.
  • Figure 4: Two step functions ($\rho_1$ and $\rho_2$) compared against a constant target distribution ($\rho^*$). The crosses on the horizontal axis represent finite agent samples drawn from each step distribution, illustrating the severe spatial displacement that occurs despite both being supported in $\mathcal{M}$, and having constant $L^2$ error.
  • Figure 5: Comparison of Euclidean ($L^2$) and Wasserstein ($W_2$) interpolations of two overlapping Gaussian distributions ($\rho_0$ and $\rho_1$). The $L^2$ interpolation ($\rho_{L^2}$) linearly joints the densities. In contrast, the $W_2$ interpolation ($\rho_W$) horizontally transports the mass, preserving the Gaussian structure.
  • Figure 6: (a), (b). Final agent positions for the regularized KDE flow with $N=1000$ agents and $u=1\cdot 10^{-3}$, $u=0.025$ respectively. (c), (d). Final $W_2(\rho^{h,N},\rho^*)$ distance with varying regularizing parameter $u$, and varying number of agents $N$.(e). Evolution of the $W_2(\rho^{h,N},\rho^*)$ distance for three different signals $u(t)$.

Theorems & Definitions (36)

  • Definition B.1
  • Definition B.2
  • Definition B.3
  • Definition B.4
  • Definition B.5
  • Definition B.6
  • Definition B.7
  • Lemma B.8
  • Definition B.9
  • Definition B.10: Comparison functions
  • ...and 26 more