CBXPy: Consensus Based Particle Dynamics in Python

CBXPy is a python package implementing consensus based particle schemes. Originally designed for optimization problems

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

the scheme was introduced as CBO (Consensus Based Optimization) in [1]. Given an ensemble of points \(x = (x_1, \ldots, x_N)\), the update reads

\[x^i \gets x^i - \lambda\, dt\, (x_i - c_\alpha(x)) + \sigma\, \sqrt{dt} |x^i - c_\alpha(x)| \xi^i\]

where \(\xi_i\) are i.i.d. standard normal random variables. The core element is the consensus point

\[c_\alpha(x) = \frac{\sum_{i=1}^N x^i\, \exp(-\alpha\, f(x^i))}{\sum_{i=1}^N \exp(-\alpha\, f(x^i))}.\]

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.

Installation

The package can be installed via pip

pip install cbx

Simple Usage Example

The following example shows how to minimize a function using CBXPy

from cbx.dynamics import CBO

f = lambda x: x[0]**2 + x[1]**2
dyn = CBO(f, d=2)
x = dyn.optimize()

More Examples

Training neural networks with CBO

A simple example showcasing CBX

Custom Noise CBX

Polarized CBO

1D Example

Visualiszing CBX dynamics

Consensus Based Sampling

Documentation

The functionality of the package is documented in the user guide. For a specifics about the implementation, we refer to the API.

• User Guide
• API Reference

References

[1]

Pinnau, R., Totzeck, C., Tse, O., & Martin, S. (2017). A consensus-based model for global optimization and its mean-field limit. Mathematical Models and Methods in Applied Sciences, 27(01), 183-204.

Indices and tables

• Index

• Module Index

• Search Page