# Convergence and error estimates for time-discrete consensus-based optimization algorithms

arXiv: https://arxiv.org/abs/2003.05086 · HTML: https://mlsanity.com/paper/2003.05086/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2020.

## Built on by (most cited first, 2 of 2)

- [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)
- [Mean-Field Model for Two-Layer Neural Networks Trained with Consensus-Based Optimization](https://mlsanity.com/paper/2511.21466/) (2025)

## Abstract (arXiv)

We present convergence and error estimates of the time-discrete consensus-based optimization(CBO) algorithms proposed in [arXiv:1909.09249] for general nonconvex functions. In authors' recent work [arxiv: 1910.08239], rigorous error analysis of the first-order consensus-based optimization algorithm proposed in [arXiv:1909.09249] was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time-discrete algorithm was not done mainly due to lack of discrete analogue of Ito's stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes the three discrete algorithms in [arXiv:1909.09249]. Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.