Consensus-based optimization with $\alpha$-stable jump processes

Federica Ferrarese; Giacomo Albi; Michael Herty; Pedro Aceves-Sanchez

arxiv: 2604.05626 · v1 · submitted 2026-04-07 · 🧮 math.OC

Consensus-based optimization with α-stable jump processes

Pedro Aceves-Sanchez , Giacomo Albi , Federica Ferrarese , Michael Herty This is my paper

Pith reviewed 2026-05-10 19:11 UTC · model grok-4.3

classification 🧮 math.OC

keywords consensus-based optimizationα-stable processesnonlocal diffusionfractional Fokker-Planck equationmean-field convergenceLévy processesstochastic optimization

0 comments

The pith

Incorporating α-stable jumps into consensus-based optimization preserves mean-field convergence while improving exploration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a variant of consensus-based optimization where particle dynamics include jumps from an α-stable stochastic process instead of standard diffusion. This produces nonlocal effects that enhance the method's ability to explore the objective landscape and escape local minima. The authors derive the corresponding fractional Fokker-Planck equation via Fourier representation and prove rigorous convergence of the particle system to the global optimizer in the mean-field limit. Numerical experiments on benchmark functions show improved performance over classical CBO, especially in multimodal or high-dimensional settings.

Core claim

By augmenting the consensus-based optimization particle dynamics with α-stable jump processes, the resulting system admits a rigorous convergence proof to the global minimum in the kinetic mean-field limit; the jumps generate a nonlocal diffusion term captured exactly by the fractional Fokker-Planck equation whose steady state concentrates on the optimizer.

What carries the argument

The α-stable jump process embedded in the kinetic particle description, which produces the nonlocal diffusion operator in the derived fractional Fokker-Planck equation.

If this is right

The method converges rigorously for a range of stability parameters α between 0 and 2.
Nonlocal diffusion improves global search capability compared with local Brownian diffusion.
The particle-level stochastic dynamics admit an exact macroscopic description via the fractional Fokker-Planck equation.
Numerical tests confirm practical gains on test functions with many local minima.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same jump mechanism could be inserted into other particle or swarm optimization schemes that rely on mean-field limits.
Long-range jumps may prove especially useful when the objective is defined on discrete or combinatorial domains.
The fractional diffusion term might interact with additional constraints or stochastic noise in the objective in ways that standard CBO does not.

Load-bearing premise

The parameters of the α-stable process can be chosen so that the nonlocal diffusion enhances exploration without disrupting the consensus mechanism or preventing the mean-field limit from existing.

What would settle it

A concrete counterexample, either analytical or numerical, in which the particles fail to reach consensus or converge to a local rather than global minimum for some admissible range of the stability parameter α.

Figures

Figures reproduced from arXiv: 2604.05626 by Federica Ferrarese, Giacomo Albi, Michael Herty, Pedro Aceves-Sanchez.

**Figure 2.** Figure 2: Modified Alpine function: three snapshots of the KB [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: Success rate and mean number of iterations for the R [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** Figure 4: Success rate and mean number of iterations for the R [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗

**Figure 5.** Figure 5: Success rate and mean number of iterations for the R [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗

**Figure 6.** Figure 6: Success rate and mean number of iterations for the d [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

**Figure 7.** Figure 7: Success rate and mean number of iterations for the d [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗

read the original abstract

In this paper, we introduce a novel variant of the CBO method that incorporates jumps according to an $\alpha$-stable stochastic process in a kinetic framework. This extension gives rise to nonlocal stochastic effects, which improve the exploration capabilities of the method. We formulate the method at the particle level, detailing the corresponding stochastic dynamics and its asymptotic behavior. In particular, through a Fourier-based representation, we derive the associated fractional Fokker-Planck equation, which naturally accounts for the nonlocal diffusion behaviors induced by $\alpha$-stable processes. As a central result, we establish a rigorous convergence result for the proposed approach. Finally, we evaluate the performance of the method through a set of numerical experiments. The results demonstrate the effectiveness of the $\alpha$-stable jump process and emphasize its potential advantages over standard diffusion-based methods, particularly in complex optimization settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper swaps Brownian motion for α-stable jumps in CBO, derives the fractional FP equation, and claims convergence, but the particle-to-mean-field step looks like the part that needs the most checking.

read the letter

The main addition is replacing the diffusion term in standard consensus-based optimization with α-stable Lévy jumps. The authors write the N-particle SDEs, use a Fourier representation to obtain the fractional Fokker-Planck equation that captures the nonlocal jumps, state a convergence theorem, and run numerical tests on benchmark problems. The numerics indicate that the jumps improve exploration on multimodal landscapes compared with the usual Brownian version, which is the practical payoff they emphasize. That extension is cleanly motivated and sits on top of the existing CBO kinetic framework without unnecessary complications. The free parameter α is left explicit, which is reasonable for a first paper on the variant. The soft spot is the mean-field passage. For 0 < α < 2 the driving process has infinite variance, so the moment-based tightness arguments that close the limit in the Brownian case do not apply directly. The Fourier step gives the formal generator for the FP equation, but the manuscript must still show that the empirical measure of the particles converges weakly to the FP solution under explicit conditions on α and the interaction kernel. If that propagation-of-chaos result is only sketched or omitted, then the convergence theorem proved on the FP level does not automatically transfer to the original stochastic system. The numerical section is standard for the field and does not raise separate issues. This work is aimed at researchers already using or analyzing particle methods for global optimization. It is worth sending to peer review because the modeling idea is straightforward and the experiments are supportive, even though referees will probably press on the limit justification and ask for clearer conditions on α.

Referee Report

2 major / 2 minor

Summary. The paper proposes a variant of consensus-based optimization (CBO) that replaces Brownian diffusion with jumps driven by an α-stable Lévy process within a kinetic (mean-field) framework. Particle-level SDEs are formulated, a Fourier-symbol representation is used to derive the associated fractional Fokker-Planck equation, a rigorous convergence theorem to the global minimizer is stated for the macroscopic equation, and numerical experiments on benchmark functions are presented to illustrate improved exploration.

Significance. If the mean-field limit and subsequent convergence result can be made fully rigorous, the work would provide a theoretically grounded way to incorporate nonlocal, heavy-tailed exploration into CBO while retaining the consensus mechanism. This could be useful for high-dimensional or multimodal optimization problems where standard diffusion-based CBO struggles with premature collapse.

major comments (2)

[Section deriving the fractional Fokker-Planck equation and the convergence theorem] The derivation of the fractional Fokker-Planck equation from the N-particle α-stable system (via Fourier representation of the generator) appears formal. For 0<α<2 the driving process has infinite variance, so standard tightness or moment-based propagation-of-chaos arguments used for Brownian CBO no longer close directly. No separate theorem is indicated that establishes weak convergence of the empirical measure to the solution of the fractional FP equation under explicit conditions on α and the interaction kernel. This step is load-bearing because the convergence theorem is proved on the FP equation rather than on the original particle system.
[Convergence theorem statement] The statement of the convergence result should clarify the precise assumptions on the objective function, the interaction kernel, the range of α, and the initial data that guarantee convergence to the global minimizer. In particular, it is unclear whether the proof adapts existing techniques for the local case or requires new estimates that control the nonlocal jumps.

minor comments (2)

Notation for the α-stable process and the fractional Laplacian should be introduced consistently between the particle SDE and the FP equation.
Numerical experiments would benefit from explicit reporting of the chosen α values, jump intensity, and how they were tuned relative to the standard CBO baseline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comments point by point below and will revise the paper to improve the rigor and clarity of the mean-field limit and convergence result.

read point-by-point responses

Referee: [Section deriving the fractional Fokker-Planck equation and the convergence theorem] The derivation of the fractional Fokker-Planck equation from the N-particle α-stable system (via Fourier representation of the generator) appears formal. For 0<α<2 the driving process has infinite variance, so standard tightness or moment-based propagation-of-chaos arguments used for Brownian CBO no longer close directly. No separate theorem is indicated that establishes weak convergence of the empirical measure to the solution of the fractional FP equation under explicit conditions on α and the interaction kernel. This step is load-bearing because the convergence theorem is proved on the FP equation rather than on the original particle system.

Authors: We agree that the derivation via Fourier symbols is formal in the current version. In the revised manuscript we will add an explicit theorem establishing weak convergence of the empirical measure to the fractional Fokker-Planck equation. The theorem will require α ∈ (1,2] to obtain sufficient integrability for tightness arguments and will impose Lipschitz and boundedness conditions on the interaction kernel. For α ≤ 1 we will add a remark on the technical obstacles posed by infinite variance. revision: yes
Referee: [Convergence theorem statement] The statement of the convergence result should clarify the precise assumptions on the objective function, the interaction kernel, the range of α, and the initial data that guarantee convergence to the global minimizer. In particular, it is unclear whether the proof adapts existing techniques for the local case or requires new estimates that control the nonlocal jumps.

Authors: We will revise the theorem statement to list all hypotheses explicitly: the objective function is C², coercive, and possesses a unique global minimizer; the interaction kernel is positive, Lipschitz continuous, and normalized; α ∈ (1,2); and the initial data has finite second moments. The proof adapts the Lyapunov-functional approach from the local CBO literature but requires additional estimates on the nonlocal fractional term; these new estimates will be outlined in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity: particle SDE to fractional FP derivation and convergence result are independent of inputs by construction

full rationale

The paper's chain proceeds from explicit particle-level SDEs with α-stable jumps, to a Fourier-symbol derivation of the fractional Fokker-Planck equation, followed by a stated rigorous convergence theorem to consensus. No quoted step reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or self-citation chain. The Fourier representation is a standard generator calculation for Lévy processes and does not presuppose the target convergence; the convergence theorem is asserted on the derived mean-field equation without evidence that it is tautological with the microscopic dynamics. This is the normal case of a self-contained derivation whose validity rests on external analysis rather than internal re-labeling of inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a well-posed mean-field limit for the particle system with α-stable jumps and on the validity of the Fourier-based representation that produces the fractional operator.

free parameters (1)

α (stability index)
The parameter controlling the jump tail heaviness; its value is chosen to balance exploration and convergence and is not derived from first principles.

axioms (1)

domain assumption The particle system admits a mean-field limit whose density satisfies the derived fractional Fokker-Planck equation.
Invoked when passing from the stochastic particle dynamics to the kinetic description.

pith-pipeline@v0.9.0 · 5443 in / 1172 out tokens · 61737 ms · 2026-05-10T19:11:04.595711+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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Abatangelo, S

N. Abatangelo, S. Dipierro, and E. Valdinoci. A Gentle Invitation to the Fractional World. Springer, 2025

work page 2025

[2] [2]

Aceves-Sanchez and L

P. Aceves-Sanchez and L. Cesbron. Fractional diﬀusion li mit for a fractional vlasov– fokker–planck equation. SIAM Journal on Mathematical Analysis , 51(1):469–488, 2019

work page 2019

[3] [3]

G. Albi, F. Ferrarese, and C. Totzeck. Kinetic-based opt imization enhanced by genetic dynamics. Mathematical Models and Methods in Applied Sciences , 33(14):2905–2933, 2023

work page 2023

[4] [4]

Albi and L

G. Albi and L. Pareschi. Binary interaction algorithms f or the simulation of ﬂocking and swarming dynamics. Multiscale Modeling & Simulation , 11(1):1–29, 2013. 28

work page 2013

[5] [5]

Ambrosio, N

L. Ambrosio, N. Gigli, and G. Savar´ e. Gradient Flows in Metric Spaces and in the Space of Probability Measures . Lectures in Mathematics ETH Z¨ urich. Birkh¨ auser Verlag, Basel, 2nd edition, 2008

work page 2008

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Applebaum

D. Applebaum. L´ evy processes and stochastic calculus. Cambridge University Press, 2009

work page 2009

[7] [7]

Benfenati, G

A. Benfenati, G. Borghi, and L. Pareschi. Binary interac tion methods for high dimen- sional global optimization and machine learning. Applied Mathematics & Optimiza- tion, 86(1):9, 2022

work page 2022

[8] [8]

Blum and A

C. Blum and A. Roli. Metaheuristics in combinatorial opt imization: Overview and conceptual comparison. ACM computing surveys (CSUR) , 35(3):268–308, 2003

work page 2003

[9] [9]

Borghi, M

G. Borghi, M. Herty, and L. Pareschi. An adaptive consens us based method for multi- objective optimization with uniform Pareto front approxim ation. Applied Mathematics & Optimization , 88(2):58, 2023

work page 2023

[10] [10]

Borghi, M

G. Borghi, M. Herty, and L. Pareschi. Kinetic models for optimization: a uniﬁed mathematical framework for metaheuristics. arXiv preprint arXiv:2410.10369 , 2024

work page arXiv 2024

[11] [11]

Borghi, H

G. Borghi, H. Im, and L. Pareschi. Swarm-based optimiza tion with jumps: a kinetic BGK framework and convergence analysis. Communications on Pure and Applied Analysis, pages 1–30, 2025

work page 2025

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J. A. Carrillo, Y.-P. Choi, C. Totzeck, and O. Tse. An ana lytical framework for consensus-based global optimization method. Mathematical Models and Methods in Applied Sciences, 28(06):1037–1066, 2018

work page 2018

[13] [13]

Dembo and O

A. Dembo and O. Zeitouni. Large deviations techniques and applications , volume 38. Springer Science & Business Media, 2009

work page 2009

[14] [14]

Modeling, Analysis, and control of Mult i- agent systems across scales

F. Ferrarese and C. Totzeck. Localized KBO with genetic dynamics for multi-modal optimization. In EMS Congress Volume “Modeling, Analysis, and control of Mult i- agent systems across scales” . AMS, 2025

work page 2025

[15] [15]

Fornasier, H

M. Fornasier, H. Huang, L. Pareschi, and P. S¨ unnen. Consensus-based optimization on hypersurfaces: Well-posedness and mean-ﬁeld limit. Mathematical Models and Methods in Applied Sciences , 30(14):2725–2751, 2020

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Fornasier, T

M. Fornasier, T. Klock, and K. Riedl. Consensus-based o ptimization methods converge globally. SIAM Journal on Optimization , 34(3):2973–3004, 2024

work page 2024

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Fornasier, L

M. Fornasier, L. Pareschi, H. Huang, and P. S¨ unnen. Con sensus-based optimization on the sphere: Convergence to global minimizers and machine learning. Journal of Machine Learning Research, 22(237):1–55, 2021. 29

work page 2021

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Fornasier and L

M. Fornasier and L. Sun. A PDE framework of consensus-ba sed optimization for objectives with multiple global minimizers. Communications in Partial Diﬀerential Equations, 50(4):493–541, 2025

work page 2025

[19] [19]

Grassi and L

S. Grassi and L. Pareschi. From particle swarm optimiza tion to consensus based optimization: stochastic modeling and mean-ﬁeld limit. Mathematical Models and Methods in Applied Sciences , 31(08):1625–1657, 2021

work page 2021

[20] [20]

Harjulehto and R

P. Harjulehto and R. Hurri-Syrj¨ anen. Pointwise estim ates to the modiﬁed Riesz po- tential. manuscripta mathematica, 156(3):521–543, 2018

work page 2018

[21] [21]

Jamil and X.-S

M. Jamil and X.-S. Yang. A literature survey of benchmar k functions for global opti- misation problems. International Journal of Mathematical Modelling and Numeri cal Optimisation, 4(2):150–194, 2013

work page 2013

[22] [22]

Kalise, A

D. Kalise, A. Sharma, and M. V. Tretyakov. Consensus-ba sed optimization via jump- diﬀusion stochastic diﬀerential equations. Mathematical Models and Methods in Ap- plied Sciences, 33(02):289–339, 2023

work page 2023

[23] [23]

Karmitsa

N. Karmitsa. Test problems for large-scale nonsmooth m inimization. Reports of the Department of Mathematical Information Technology. Series B, Scientiﬁc computing , (4/2007), 2007

work page 2007

[24] [24]

S. Ken-Iti. L´ evy processes and inﬁnitely divisible distributions, volume 68. Cambridge University Press, 1999

work page 1999

[25] [25]

Kennedy and R

J. Kennedy and R. Eberhart. Particle swarm optimizatio n. In Proceedings of ICNN’95- international conference on neural networks , volume 4, pages 1942–1948. ieee, 1995

work page 1942

[26] [26]

A. E. Kyprianou and J. C. Pardo. Stable L´ evy processes via Lamperti-type represen- tations, volume 7. Cambridge University Press, 2022

work page 2022

[27] [27]

M. M. Meerschaert and A. Sikorskii. Stochastic models for fractional calculus , vol- ume 43. Walter de Gruyter GmbH & Co KG, 2019

work page 2019

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Michalewicz

Z. Michalewicz. Genetic algorithms+ data structures= evolution programs . springer science & business media, 1996

work page 1996

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P. D. Miller. Applied asymptotic analysis , volume 75. American Mathematical Soc., 2006

work page 2006

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Mitchell

M. Mitchell. Genetic algorithms: An overview. In Complex., volume 1, pages 31–39, 1995

work page 1995

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K. Nanbu. Direct simulation scheme derived from the Bol tzmann equation. I. Mono- component gases. Journal of the Physical Society of Japan , 49(5):2042–2049, 1980. 30

work page 2042

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Pareschi and G

L. Pareschi and G. Russo. An introduction to Monte Carlo method for the Boltzmann equation. In ESAIM: Proceedings, volume 10, pages 35–75. EDP Sciences, 2001

work page 2001

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Pareschi and G

L. Pareschi and G. Toscani. Interacting multiagent systems: kinetic equations and Monte Carlo methods . OUP Oxford, 2013

work page 2013

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Pinnau, C

R. Pinnau, C. Totzeck, O. Tse, and S. Martin. A consensus -based model for global optimization and its mean-ﬁeld limit. Mathematical Models and Methods in Applied Sciences, 27(01):183–204, 2017. 31

work page 2017