# Zero-Inertia Limit: from Particle Swarm Optimization to Consensus Based Optimization

arXiv: https://arxiv.org/abs/2104.06939 · HTML: https://mlsanity.com/paper/2104.06939/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2021.

## Built on by (most cited first, 1 of 1)

- [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)

## Abstract (arXiv)

Recently a continuous description of the particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in arXiv:2012.05613 where the authors formally showed the link between PSO and the consensus based optimization (CBO) through zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in arXiv:2012.05613 by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space and the results are illustrated with some numerical examples.