Convergence Error Analysis of Reflected Gradient Langevin Dynamics for Globally Optimizing Non-Convex Constrained Problems

TL;DR

This paper analyzes the convergence rates of reflected gradient Langevin dynamics for globally optimizing non-convex problems with non-convex constraints, demonstrating faster convergence than previous methods.

Contribution

It extends gradient Langevin dynamics to non-convex constrained problems using reflections and derives improved convergence rates.

Findings

Faster convergence rates than existing methods for convex constrained non-convex problems.

Effective use of boundary reflection enhances the optimization process.

Probabilistic representation aids in analyzing convergence behavior.

Abstract

Gradient Langevin dynamics and a variety of its variants have attracted increasing attention owing to their convergence towards the global optimal solution, initially in the unconstrained convex framework while recently even in convex constrained non-convex problems. In the present work, we extend those frameworks to non-convex problems on a non-convex feasible region with a global optimization algorithm built upon reflected gradient Langevin dynamics and derive its convergence rates. By effectively making use of its reflection at the boundary in combination with the probabilistic representation for the Poisson equation with the Neumann boundary condition, we present promising convergence rates, particularly faster than the existing one for convex constrained non-convex problems.