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

Minimizing a function using CBXPy can be done as follows:

   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.

Currently CBXPy can only be installed from PyPI with pip.

   pip install 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 $x = (x_1, \ldots, x_N)$, the update reads

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

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

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

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.

Among others, CBXPy currently implments

• CBO (Consensus-Based Optimization) [1],
• CBS (Consensus-Based Sampling) [2],
• CBO with memory [3],
• Batching schemes [4].

[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