cbx 0.1.1

CBXpy

What is CBXPy?

CBXPy is 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

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

A Documentation is available at https://pdips.github.io/CBXpy

Installation

What is CBX?

Originally designed for optimization problems

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

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 variables. 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.

Usage examples