Wasserstein Contraction Bounds on Closed Convex Domains with Applications to Stochastic Adaptive Control

TL;DR

This paper extends Wasserstein contraction bounds to constrained stochastic systems, enabling quantitative analysis of adaptive control convergence and stability in reflected SDEs with practical applications.

Contribution

It generalizes contraction theory from unconstrained to constrained reflected SDEs, providing new bounds for stochastic adaptive control systems.

Findings

Derived contraction bounds for reflected SDEs.

Applied bounds to nonlinear stochastic adaptive regulation.

Extended theory to accommodate constrained parameter spaces.

Abstract

This paper is motivated by the problem of quantitatively bounding the convergence of adaptive control methods for stochastic systems to a stationary distribution. Such bounds are useful for analyzing statistics of trajectories and determining appropriate step sizes for simulations. To this end, we extend a methodology from (unconstrained) stochastic differential equations (SDEs) which provides contractions in a specially chosen Wasserstein distance. This theory focuses on unconstrained SDEs with fairly restrictive assumptions on the drift terms. Typical adaptive control schemes place constraints on the learned parameters and their update rules violate the drift conditions. To this end, we extend the contraction theory to the case of constrained systems represented by reflected stochastic differential equations and generalize the allowable drifts. We show how the general theory can be used to derive quantitative contraction bounds on a nonlinear stochastic adaptive regulation problem.