Consensus-Based Optimization Methods Converge Globally

TL;DR

This paper provides a rigorous theoretical analysis of consensus-based optimization (CBO), demonstrating its global convergence properties and unveiling its internal mechanisms, with implications for solving complex nonconvex optimization problems.

Contribution

It introduces a novel convergence proof for CBO in mean-field law, showing convexification effects and improving assumptions for broader applicability.

Findings

CBO performs a gradient descent on average towards the global minimizer.

The method convexifies a wide class of optimization problems as the number of agents increases.

Probabilistic convergence guarantees for the numerical CBO method are established.

Abstract

In this paper, we study consensus-based optimization (CBO), which is a multi-agent metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. Based on an experimentally supported intuition that, on average, CBO performs a gradient descent of the squared Euclidean distance to the global minimizer, we devise a novel technique for proving the convergence to the global minimizer in mean-field law for a rich class of objective functions. The result unveils internal mechanisms of CBO that are responsible for the success of the method. In particular, we prove that CBO performs a convexification of a large class of optimization problems as the number of optimizing agents goes to infinity. Furthermore, we improve prior analyses by requiring mild assumptions about the initialization of the method and by covering objectives that are merely locally Lipschitz continuous. As a core component of this analysis, we establish a quantitative nonasymptotic Laplace principle, which may be of independent interest. From the result of CBO convergence in mean-field law, it becomes apparent that the hardness of any global optimization problem is necessarily encoded in the rate of the mean-field approximation, for which we provide a novel probabilistic quantitative estimate. The combination of these results allows to obtain probabilistic global convergence guarantees of the numerical CBO method.