Stochastic monotone inclusion with closed loop distributions
Résumé
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.
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