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Stochastic Monotone Inclusion with Closed Loop Distributions

Hamza Ennaji, Jalal Fadili, Hedy Attouch

TL;DR

This work develops a rigorous framework for stochastic monotone inclusions with decision-dependent distributions in a Hilbert space, framing equilibria as zeros of $A+B_{ar{x}}$ and linking the Lipschitz sensitivity of the distribution in $\mathds{W}_1$ to coarse Ollivier-Ricci curvature. It analyzes both first-order and inertial second-order dynamics with Lipschitz perturbations, establishing existence, uniqueness, and convergence rates under monotonicity and curvature conditions. A key contribution is the application to inertial primal-dual algorithms for composite stochastic objectives, showing well-posedness and exponential convergence of the primal-dual trajectories to a unique equilibrium. The results illuminate how curvature-like conditions govern stability and convergence in stochastic, decision-dependent settings and suggest practical inertial schemes with Hessian damping for efficient optimization under uncertainty.

Abstract

In this paper, we study in a Hilbertian setting, first and second-order monotone inclusions related to stochastic optimization problems with decision dependent distributions. The studied dynamics are formulated as monotone inclusions governed by Lipschitz perturbations of maximally monotone operators where the concept of equilibrium plays a central role. We discuss the relationship between the $\mathbb{W}_1$-Wasserstein Lipschitz behavior of the distribution and the so-called coarse Ricci curvature. As an application, we consider the monotone inclusions associated with stochastic optimisation problems involving the sum of a smooth function with Lipschitz gradient, a proximable function and a composite term.

Stochastic Monotone Inclusion with Closed Loop Distributions

TL;DR

This work develops a rigorous framework for stochastic monotone inclusions with decision-dependent distributions in a Hilbert space, framing equilibria as zeros of and linking the Lipschitz sensitivity of the distribution in to coarse Ollivier-Ricci curvature. It analyzes both first-order and inertial second-order dynamics with Lipschitz perturbations, establishing existence, uniqueness, and convergence rates under monotonicity and curvature conditions. A key contribution is the application to inertial primal-dual algorithms for composite stochastic objectives, showing well-posedness and exponential convergence of the primal-dual trajectories to a unique equilibrium. The results illuminate how curvature-like conditions govern stability and convergence in stochastic, decision-dependent settings and suggest practical inertial schemes with Hessian damping for efficient optimization under uncertainty.

Abstract

In this paper, we study in a Hilbertian setting, first and second-order monotone inclusions related to stochastic optimization problems with decision dependent distributions. The studied dynamics are formulated as monotone inclusions governed by Lipschitz perturbations of maximally monotone operators where the concept of equilibrium plays a central role. We discuss the relationship between the -Wasserstein Lipschitz behavior of the distribution and the so-called coarse Ricci curvature. As an application, we consider the monotone inclusions associated with stochastic optimisation problems involving the sum of a smooth function with Lipschitz gradient, a proximable function and a composite term.
Paper Structure (30 sections, 28 theorems, 147 equations, 1 figure, 1 table)

This paper contains 30 sections, 28 theorems, 147 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Contributions and organization of the paper
  4. Notation and preliminaries
  5. Convex analysis
  6. Operator theory
  7. Monotone inclusions
  8. Transportation distance
  9. First-order monotone inclusions
  10. Existence and uniqueness of equilibria
  11. Well-posedness
  12. Convergence properties
  13. Inertial second-order system with viscous and Hessian damping
  14. Well-posedness
  15. Equivalent first-order formulation
  16. ...and 15 more sections

Key Result

Theorem 1

Under assump:4, the map is a contraction. In particular, the equilibrium $\bar{x}$ is unique. If moreover, assump:3 holds instead, i.e., $F_{\mathsf{m}_{x}}$ is $\mu$-strongly monotone for $\mu>0$, the mapping $S$ is $\rho$-Lipschitz with $\rho:=\frac{\beta\tau}{\mu}$. Thus for $\rho < 1$, there exists a unique equilibrium

Figures (1)

  • Figure 1: Illustration of the ORC.

Theorems & Definitions (65)

  • Definition 1: Differentiability
  • Definition 2: $L$-smoothness
  • Definition 3
  • Remark 2.1
  • Definition 4
  • Definition 5
  • Theorem 1: Existence and uniqueness of equilibrium point
  • proof
  • Remark 3.1
  • Lemma 1
  • ...and 55 more