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| --- | ||
| title: 'CBX: Python and Julia packages of consensus-based interacting particle methods' | ||
| title: 'CBX: Python and Julia packages for consensus-based interacting particle methods' | ||
| tags: | ||
| - Python | ||
| - Sampling | ||
| - Julia | ||
| - Optimization | ||
| - Sampling | ||
| authors: | ||
| - name: Tim Roith | ||
| orcid: 0000-0001-8440-2928 | ||
| affiliation: 1 | ||
| - name: Konstantin Riedl | ||
| orcid: 0000-0002-2206-4334 | ||
| affiliation: "2, 3" | ||
| affiliations: | ||
| - name: Friedrich-Alexander-Universität Erlangen-Nürnberg | ||
| index: 1 | ||
| date: 23 August 2023 | ||
| - name: Technical University of Munich | ||
| index: 2 | ||
| - name: Munich Center for Machine Learning | ||
| index: 3 | ||
| date: 08 October 2023 | ||
| bibliography: paper.bib | ||
| --- | ||
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| # Summary | ||
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| We present CBXpy and CBX.jl which provide Python and respectively Julia implementations for consensus-based interacting particle methods. In detail, the packages focus on consensus-based optimization (CBO) [@pinnau2017consensus] and consensus-based sampling (CBS) [@carrillo2022consensus], which coined the acronym CBX. The Python and Julia implementations were developed in parallel, in order to provide a framework for researchers more familiar with either language. Here, we focused on having a similar API and core functionality in both packages, while taking advantage of the strengths of each language, and writing idiomatic code. | ||
| We present CBXpy and CBX.jl which provide Python and Julia implementations, respectively, for consensus-based interacting particle methods. In detail, the packages focus on consensus-based optimization (CBO) [@pinnau2017consensus] and consensus-based sampling (CBS) [@carrillo2022consensus], which coined the acronym CBX. The Python and Julia implementations were developed in parallel, in order to provide a framework for researchers more familiar with either language. While we focused on having a similar API and core functionality in both packages, we took advantage of the strengths of each language and wrote idiomatic code. | ||
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| ![Visualization of a CBO run for the Ackley function [@ackley2012connectionist].](JOSS.png){ width=50% } | ||
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| # Statement of need | ||
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| Consensus-based optimization (CBO) was proposed in [@pinnau2017consensus] as a zeroth-order particle-based scheme, to solve problems of the form | ||
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| Consensus-based optimization (CBO) was proposed in [@pinnau2017consensus] as a zero-order (derivative-free) particle-based scheme, to solve problems of the form | ||
| $$ | ||
| min_{x\in\mathcal{X}} f(x) | ||
| x^* = \mathrm{argmin}_{x\in\mathcal{X}} f(x), | ||
| $$ | ||
| for some input space $\mathcal{X}$ and a possibly nonconvex and nonsmooth objective function $f:\mathcal{X}\to\mathbb{R}$. As an agent-based method, CBO is conceptually comparable to biologically and physically inspired methods such as particle-swarm optimization (PSO) [@kennedy1995particle], simulated annealing (SA) [@henderson2003theory] or several other heuristics [@mohan2012survey;@karaboga2014comprehensive;@yang2009firefly;@bayraktar2013wind]. However, compared to these methods, CBO was designed to be amenable to a rigorous theoretical convergence analysis [@carrillo2018analytical;@carrillo2021consensus;@fornasier2021consensus;@fornasier2021convergence]. From a computational perspective, the method is attractive as the particle interactions scale linearly with the number of particles. | ||
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| for some input space $\mathcal{X}$ and a possibly non-convex objective function $f:\mathcal{X}\to\mathbb{R}$. As an agent-based method, CBO is conceptually comparable to biologically and physically inspired methods [@mohan2012survey;karaboga2014comprehensive;yang2009firefly;bayraktar2013wind], particle-swarm optimization (PSO) [@kennedy1995particle] or simulated annealing [@henderson2003theory]. However, compared to other heuristics, one can derive a limiting PDE in the infinite-particle limit, which has sparked considerable theoretical interest in recent years [@totzeck2021trends]. From a computational side, the method is also attractive, since the amount of particle interaction scales linearly with the number of particles. | ||
| For Python, PSO and SA implementations are already available [@miranda2018pyswarms;@scikitopt;@deapJMLR2012;@pagmo2017], which are widely used in the community and provide a rich framework for the respective methods. However, adjusting these implementations to CBO is not straightforward. Furthermore, in this project, we want to provide a lightweight and direct implementation of CBO methods, which are easy to understand and to modify. The first publicly available Python packages implementing CBO-type algorithms were given by some of the authors together with collaborators in [@Igor_CBOinPython], where CBO as in [@pinnau2017consensus] is implemented, as well as in [@Roith_polarcbo], where so-called polarized CBO [@bungert2022polarized] is implemented. The current Python package is a complete rewrite of the latter implementation. | ||
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| For Python, PSO and SA implementations are already available [@miranda2018pyswarms;@scikitopt;@deapJMLR2012;@pagmo2017], which are widely used in the community and provide a rich framework for the respective methods. However, adjusting these implementations to CBO is not straightforward. Furthermore, in this project we want to provide a lightweight and direct implementation of the method, which is easy to understand and modify. The first publicly available python package implementing CBO type algorithms was given by one of the authors in [@Roith_polarcbo] implementing so-called polarized CBO [@bungert2022polarized]. The current package is a complete rewrite of this previous implementation. | ||
| For Julia, PSO and SA methods are, among others, implemented in [@mogensen2018optim;@mejia2022metaheuristics;@Bergmann2022]. Similarly, one of the authors provided the first specific Julia implementation of CBO [@Bailo_consensus]. However, the current version of the package deviates from the previous implementation and is more closely oriented toward the Python implementation. | ||
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| For Julia PSO and SA methods are among others impemented in [@mogensen2018optim;mejia2022metaheuristics;Bergmann2022]. Similarly, one of the authors provided the first specific Julia implementation of CBO [@Bailo_consensus]. However, the current version of the package deviates from the previous implementation and is more closely related to the Python implementation. | ||
| We summarize the motivation and main features of the packages in what follows. | ||
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| We summarizes the motivation and main features of the packages in the following: | ||
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| - Provide a lightweight and easy to understand implementation of CBO and variants such as batched [@carrillo2021consensus] or polarized CBO [@bungert2022polarized]. The implementation relie | ||
| - Provide a lightweight, easy-to-understand, -use and -extend implementation of CBO together with several of its variants. These include CBO with mini-batching [@carrillo2021consensus], polarized CBO [@bungert2022polarized], CBO with memory effects [@grassi2020particle;@riedl2022leveraging], and CBS [@carrillo2022consensus]. The implementation relies on ... | ||
| - torch and tensorflow-like usage style (and implementation way via step), maybe provide here code snippet | ||
| - | ||
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| # Mathematical background | ||
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| CBO methods use a finite number of agents $X^1,\dots,X^N$ to explore the domain and to form a global consensus about the location of the minimizer $x^*$ as time passes. They are described through a system of stochastic differential equations (SDEs), expressed in It\^o's form as | ||
| $$ | ||
| dX^i_t = -\lambda (X^i_t-x_\alpha(\widehat\rho_t^N)) dt + \sigma D(X^i_t-x_\alpha(\widehat\rho_t^N)) dB^i_t, | ||
| $$ | ||
| where $\alpha,\lambda$ and $\sigma$ are user-specified parameters, $((B^i_t)_{t\geq0})_{i=1,\dots,N}$ denote independent standard Brownian motions and where $x_\alpha(\widehat\rho_t^N)$ denotes the consensus point, a suitably weighted average of the positions of the particles, which is computed as | ||
| $$ | ||
| x_\alpha(\widehat\rho_t^N) = \frac{1}{\sum_{i=1}^N \omega_\alpha(X^i_t)} \sum_{i=1}^N X^i_t\omega_\alpha(X^i_t), \quad\text{ with }\quad \omega_\alpha(x) = \mathrm{exp}(-\alpha f(x)). | ||
| $$ | ||
| A theoretical convergence analysis is not directly conducted on the above SDE system due its highly complex behavior, but on its macroscopic mean-field limit (infinite-particle limit) [@huang2021MFLCBO], which can be described by a nonlinear nonlocal Fokker-Planck equation [@pinnau2017consensus;@carrillo2018analytical;@carrillo2021consensus;@fornasier2021consensus;@fornasier2021convergence]. The implemented CBO code originates from a simple Euler-Maruyama time discretization of the above SDE system. A convergence statement therefore is available in [@fornasier2021consensus]. | ||
| Similar analysis techniques further allowed to obtain theoretical convergence guarantees for PSO [@qiu2022PSOconvergence]. | ||
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| # Application areas of CBX | ||
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| As of now, CBX methods have been deployed in a variety of different settings and for different purposes, such as for solving constrained optimizations [@fornasier2020consensus_sphere_convergence;@borghi2021constrained], multi-objective optimizations [@borghi2022adaptive;@klamroth2022consensus], saddle point problems [@huang2022consensus], federated learning tasks [@carrillo2023fedcbo], adversarial training [] or for sampling [@carrillo2022consensus]. | ||
| In addition, recent work [@riedl2023gradient] establishes a connection of CBO to stochastic gradient descent-type methods, suggesting a more fundamental connection of theoretical interest between derivative-free and gradient-based methods. | ||
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| # Acknowledgements | ||
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| mention Lorentz Centre in Leiden | ||
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| # References |