Stochastic Monotone Inclusion with Closed Loop Distributions

TL;DR

This paper investigates stochastic monotone inclusions in a Hilbert space setting, exploring their properties and relationships with distributional Lipschitz behavior, and applies these findings to stochastic optimization problems with complex structures.

Contribution

It introduces a novel framework linking stochastic monotone inclusions with distributional Lipschitz properties and equilibrium concepts, advancing understanding in stochastic optimization.

Findings

Established connections between Wasserstein Lipschitz behavior and coarse Ricci curvature.

Analyzed dynamics governed by Lipschitz perturbations of monotone operators.

Applied theoretical results to complex stochastic optimization problems.

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.