# Consensus-Based Optimization Methods Converge Globally

arXiv: https://arxiv.org/abs/2103.15130 · HTML: https://mlsanity.com/paper/2103.15130/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Mar 2021.

## Built on by (most cited first, 16 of 16)

- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Stochastic Zeroth order Descent with Structured Directions](https://mlsanity.com/paper/2206.05124/) (2022)
- [FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization](https://mlsanity.com/paper/2305.02894/) (2023)
- [Adaptive State-Dependent Diffusion for Derivative-Free Optimization](https://mlsanity.com/paper/2302.04370/) (2023)
- [Quantifying Concentration Phenomena of Mean-Field Transformers in the Low-Temperature Regime](https://mlsanity.com/paper/2605.10931/) (2026)
- [Gradient is All You Need? How Consensus-Based Optimization can be Interpreted as a Stochastic Relaxation of Gradient Descent](https://mlsanity.com/paper/2306.09778/) (2023)
- [Consensus-based optimization for closed-box adversarial attacks and a connection to evolution strategies](https://mlsanity.com/paper/2506.24048/) (2025)
- [Defending Against Diverse Attacks in Federated Learning Through Consensus-Based Bi-Level Optimization](https://mlsanity.com/paper/2412.02535/) (2024)
- [Global Optimization via Softmin Energy Minimization](https://mlsanity.com/paper/2509.17815/) (2025)
- [Mean-Field Model for Two-Layer Neural Networks Trained with Consensus-Based Optimization](https://mlsanity.com/paper/2511.21466/) (2025)
- [Ensemble-based gradient inference for particle methods in optimization and sampling](https://mlsanity.com/paper/2209.15420/) (2022)
- [Non-intrusive surrogate modelling using sparse random features with applications in crashworthiness analysis](https://mlsanity.com/paper/2212.14507/) (2022)
- [Consensus Based Stochastic Control](https://mlsanity.com/paper/2501.17801/) (2025)
- [A Structured Tour of Optimization with Finite Differences](https://mlsanity.com/paper/2505.19720/) (2025)
- [Proximal basin hopping: global optimization with guarantees](https://mlsanity.com/paper/2605.18364/) (2026)
- [Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees](https://mlsanity.com/paper/2606.14289/) (2026)

## Abstract (arXiv)

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.