# Convergence of Anisotropic Consensus-Based Optimization in Mean-Field Law

arXiv: https://arxiv.org/abs/2111.08136 · HTML: https://mlsanity.com/paper/2111.08136/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2021.

## Built on by (most cited first, 5 of 5)

- [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)
- [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)

## Abstract (arXiv)

In this paper we study anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method capable of globally minimizing nonconvex and nonsmooth functions in high dimensions. CBO is based on stochastic swarm intelligence, and inspired by consensus dynamics and opinion formation. Compared to other metaheuristic algorithms like particle swarm optimization, CBO is of a simpler nature and therefore more amenable to theoretical analysis. By adapting a recently established proof technique, we show that anisotropic CBO converges globally with a dimension-independent rate for a rich class of objective functions under minimal assumptions on the initialization of the method. Moreover, the proof technique reveals that CBO performs a convexification of the optimization problem as the number of agents goes to infinity, thus providing an insight into the internal CBO mechanisms responsible for the success of the method. To motivate anisotropic CBO from a practical perspective, we further test the method on a complicated high-dimensional benchmark problem, which is well understood in the machine learning literature.