A Python package for consensus-based particle dynamics, focusing on optimization and sampling.

How to use CBXPy?

Minimizing a function using CBXPy can be done as follows:

```python from cbx.dynamics import CBO # import the CBO class

f = lambda x: x[0]2 + x[1]2 # define the function to minimize x = CBO(f, d=2).optimize() # run the optimization ```

A documentation together with more examples and usage instructions is available at https://pdips.github.io/CBXpy.

Installation

Currently CBXPy can only be installed from PyPI with pip.

bash pip install cbx

What is CBX?

Originally designed for optimization problems of the form

$$ \min_{x \in \mathbb{R}^n} f(x), $$

the scheme was introduced as CBO (Consensus-Based Optimization). Given an ensemble of points $x = (x1, \ldots, xN)$, the update reads

$$ xi \gets xi - \lambda\ dt\ (xi - c(x)) + \sigma\ \sqrt{dt} |xi - c(x)|\ \xi_i $$

where $\xi_i$ are i.i.d. standard normal random vectors. The core element is the consensus point

$$ \begin{align} c(x) = \left(\sum{i=1}^{N} xi\ \exp(-\alpha\ f(xi))\right)\bigg/\left(\sum{i=1}^N \exp(-\alpha\ f(x_i))\right) \end{align} $$

with a parameter $\alpha>0$. The scheme can be extended to sampling problems known as CBS, clustering problems and opinion dynamics, which motivates the acronym CBX, indicating the flexibility of the scheme.

Functionality

Among others, CBXPy currently implements

• CBO (Consensus-Based Optimization) [1]
• CBS (Consensus-Based Sampling) [2]
• CBO with memory [3]
• Batching schemes [4]
• Polarized CBO [5]
• Mirror CBO [6]
• Adamized CBO [7]
• Constrained CBO methods, including
  • Drift Correction [8]
  • Regularization [9]
  • Hypersurface CBO [10]

References

[1] A consensus-based model for global optimization and its mean-field limit, Pinnau, R., Totzeck, C., Tse, O. and Martin, S., Mathematical Models and Methods in Applied Sciences 2017

[2] Consensus-based sampling, Carrillo, J.A., Hoffmann, F., Stuart, A.M., and Vaes, U., Studies in Applied Mathematics 2022

[3] Leveraging Memory Effects and Gradient Information in Consensus-Based Optimization: On Global Convergence in Mean-Field Law, Riedl, K., 2022

[4] A consensus-based global optimization method for high dimensional machine learning problems, Carrillo, J.A., Jin, S., Li, L. and Zhu, Y., ESAIM: Control, Optimisation and Calculus of Variations 2021

[5] Bungert, L., Roith, T., & Wacker, P. (2024). Polarized consensus-based dynamics for optimization and sampling. Mathematical Programming, 1-31.

[6] Bungert, L., Hoffmann, F., Kim, D. Y., & Roith, T. (2025). MirrorCBO: A consensus-based optimization method in the spirit of mirror descent. arXiv preprint arXiv:2501.12189.

[7] Chen, J., Jin, S., & Lyu, L. (2020). A consensus-based global optimization method with adaptive momentum estimation. arXiv preprint arXiv:2012.04827.

[8] Carrillo, J. A., Jin, S., Zhang, H., & Zhu, Y. (2024). An interacting particle consensus method for constrained global optimization. arXiv preprint arXiv:2405.00891.

[9] Borghi, G., Herty, M., & Pareschi, L. (2023). Constrained consensus-based optimization. SIAM Journal on Optimization, 33(1), 211-236.

[10] Fornasier, M., Huang, H., Pareschi, L., & Sünnen, P. (2020). Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit. Mathematical Models and Methods in Applied Sciences, 30(14), 2725-2751.