Consensus-Based Optimization with Truncated Noise

Konstantin Riedl; Lukang Sun; Massimo Fornasier; Peter Richt\'arik

arxiv: 2310.16610 · v2 · submitted 2023-10-25 · 🧮 math.OC

Consensus-Based Optimization with Truncated Noise

Massimo Fornasier , Peter Richt\'arik , Konstantin Riedl , Lukang Sun This is my paper

Pith reviewed 2026-05-24 06:44 UTC · model grok-4.3

classification 🧮 math.OC

keywords consensus-based optimizationtruncated noiseglobal optimizationWasserstein distanceparticle systemsconvergence in expectationnonconvex optimization

0 comments

The pith

Truncating the noise term in consensus-based optimization bounds higher moments of the particle law and proves convergence in expectation to the global minimizer under minimal assumptions.

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

The paper modifies the standard consensus-based optimization method by capping the noise added to each particle's velocity. This change keeps higher moments of the empirical particle measure from blowing up, which was a limitation in the untruncated version. With those bounds in hand, the authors track the time evolution of the Wasserstein-2 distance between the particle measure and the global minimizer. Under only minimal conditions on the objective function and the starting positions, they obtain convergence in expectation. Experiments indicate that the truncated variant tolerates a wider range of noise strengths while still converging reliably.

Core claim

By replacing the unbounded noise in the CBO particle dynamics with a truncated version, the law of the interacting particles satisfies uniform bounds on moments of all orders. These bounds close the estimates needed to control the evolution of the Wasserstein-2 distance to the global minimizer, yielding convergence in expectation of the empirical measure with only minimal assumptions on the objective and the initial configuration.

What carries the argument

The truncated noise term added to each particle's drift, which caps the Gaussian increment so that the resulting measure satisfies moment bounds sufficient to close the Wasserstein-2 contraction argument.

If this is right

The method converges in expectation for a larger set of nonconvex and nonsmooth objectives than the original CBO formulation.
A wider interval of admissible noise strengths becomes usable without losing the convergence guarantee.
Higher-moment control removes the need for additional regularization or projection steps that were sometimes required before.
The same truncation technique can be inserted into other multi-particle metaheuristics whose analysis previously relied on moment bounds.

Where Pith is reading between the lines

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

The truncation idea may transfer to other interacting-particle schemes used for sampling or sampling-based optimization.
Because the proof relies only on Wasserstein-2 contraction, similar arguments could be attempted with other optimal-transport distances or with mean-field limits.
Practical implementations could adapt the truncation threshold dynamically rather than keeping it fixed.

Load-bearing premise

The objective function and the initial particle configuration satisfy the minimal conditions that let the Wasserstein-2 distance evolution be controlled and produce convergence in expectation.

What would settle it

An objective function and initial particle configuration satisfying the stated minimal conditions for which the truncated-noise system fails to drive the Wasserstein-2 distance to zero in expectation.

Figures

Figures reproduced from arXiv: 2310.16610 by Konstantin Riedl, Lukang Sun, Massimo Fornasier, Peter Richt\'arik.

**Figure 1.** Figure 1: A comparison of the success probabilities of isotropic CBO with (left phase diagrams) and without (right separate columns) truncated noise for different values of the truncation parameter M and the noise level σ. (Note that standard CBO as investigated in [11, 23, 40] is retrieved when choosing M = ∞, R = ∞ and vb = 0 in (1)). In both settings (a) and (b) the depicted success probabilities are averaged ove… view at source ↗

read the original abstract

Consensus-based optimization (CBO) is a versatile multi-particle metaheuristic optimization method suitable for performing nonconvex and nonsmooth global optimizations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analyzing the time-evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimizer of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialization. Numerical evidences demonstrate the benefit of truncating the noise in CBO.

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

Truncating the noise in CBO bounds higher moments, widens usable noise parameters, and supports a W2 convergence proof under minimal assumptions.

read the letter

The main takeaway is that adding truncation to the noise term in the CBO particle system prevents higher moments from blowing up. This lets the authors bound the statistics more cleanly than in the original formulation and prove that the empirical measure converges in expectation to the global minimizer by tracking the Wasserstein-2 distance evolution. The truncation also permits larger noise values in experiments without destabilizing the dynamics. The change is simple and directly targets a known practical weakness in standard CBO. The analysis stays focused on the distance to the minimizer rather than deriving everything from a mean-field limit first, which keeps the argument relatively direct. Experiments are cited to show the benefit on parameter flexibility. The soft spots are modest but real. The abstract labels the assumptions “minimal” without listing them, so it is not yet clear how much weaker they are than prior CBO results or whether the truncation is required to close the differential inequality for the W2 moment. The numerical section is described only at a high level, leaving open questions about the range of test problems and the size of the observed gains. No circularity or hidden fitting appears in the stated argument structure. This work is aimed at people who already use or analyze consensus-based or particle methods for nonconvex global optimization. A reader who cares about moment control and provable convergence for these heuristics will find the technical step useful. It deserves peer review because the core modification is concrete, the claim is specific, and the proof outline is checkable.

Referee Report

1 major / 0 minor

Summary. The paper proposes a truncated-noise variant of consensus-based optimization (CBO) that bounds higher moments of the particle system law, thereby improving well-behavedness and allowing greater flexibility in the noise parameter. Convergence in expectation is proved by deriving a differential inequality for the Wasserstein-2 distance between the empirical measure and the global minimizer, under minimal assumptions on the objective and initialization; numerical experiments support the practical benefits.

Significance. If the stated convergence result holds, the work strengthens the theoretical toolkit for CBO by supplying explicit moment control via truncation and a proof that avoids strong regularity requirements on the objective, which could widen the range of admissible noise intensities in applications.

major comments (1)

[Abstract / Theorem statement] The abstract asserts convergence 'requiring only minimal assumptions on the objective function and on the initialization,' yet neither the abstract nor the provided description lists these assumptions explicitly. Because the entire proof strategy rests on controlling the W2 evolution under precisely these conditions, their precise statement (e.g., growth, coercivity, or initialization moment hypotheses) must appear before the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment. We address the single major comment below.

read point-by-point responses

Referee: [Abstract / Theorem statement] The abstract asserts convergence 'requiring only minimal assumptions on the objective function and on the initialization,' yet neither the abstract nor the provided description lists these assumptions explicitly. Because the entire proof strategy rests on controlling the W2 evolution under precisely these conditions, their precise statement (e.g., growth, coercivity, or initialization moment hypotheses) must appear before the main theorem.

Authors: We agree that the abstract does not enumerate the assumptions and that doing so would improve readability. The assumptions (unique global minimizer, coercivity at infinity with quadratic growth, and finite second-moment initialization) are stated explicitly in Assumption 2.1 and restated immediately before Theorem 3.1. To address the referee's concern we will revise the abstract to include a concise bullet list of these hypotheses and will add a short paragraph immediately preceding the theorem that recalls them verbatim. This change does not alter the proof but makes the precise conditions under which the W2 analysis holds transparent from the outset. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result is a direct analysis of the Wasserstein-2 distance evolution for the truncated-noise CBO dynamics, yielding an expectation convergence bound under minimal assumptions on f and the initial measure. This derivation does not reduce any quantity to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the truncation is introduced explicitly to close the moment bounds needed for the differential inequality, and the proof proceeds from the SDE to the W2 evolution without importing uniqueness or ansatzes from prior author work as an external forcing step. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5730 in / 1003 out tokens · 20200 ms · 2026-05-24T06:44:40.934224+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

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Aarts and J

E. Aarts and J. Korst. Simulated annealing and Boltzmann machines. A stochastic approach to combinatorial optimization and neural computing . Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Ltd., Chichester, 1989

work page 1989

[2] [2]

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, second edition, 2008

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H.-O. Bae, S.-Y. Ha, M. Kang, H. Lim, C. Min, and J. Yoo. A constrained consensus based optimization algorithm and its application to finance. Appl. Math. Comput. , 416:Paper No. 126726, 10, 2022

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G. Borghi, S. Grassi, and L. Pareschi. Consensus based optimization with memory effects: random selection and applications. arXiv preprint arXiv:2301.13242 , 2023

work page arXiv 2023

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Borghi, M

G. Borghi, M. Herty, and L. Pareschi. A consensus-based algorithm for multi-objective optimization and its mean-field description. In 2022 IEEE 61st Conference on Decision and Control (CDC) , pages 4131–4136. IEEE, 2022

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Borghi, M

G. Borghi, M. Herty, and L. Pareschi. An adaptive consensus based method for multi- objective optimization with uniform Pareto front approximation. Applied Mathematics & Optimization, 88(2):1–43, 2023

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G. Borghi, M. Herty, and L. Pareschi. Constrained consensus-based optimization. SIAM Journal on Optimization , 33(1):211–236, 2023

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work page arXiv 2023

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C. Cipriani, H. Huang, and J. Qiu. Zero-inertia limit: from particle swarm optimization to consensus-based optimization. SIAM J. Appl. Math , 54(3):3091–3121, 2022

work page 2022

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Fornasier, H

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

work page 2020

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Fornasier, H

M. Fornasier, H. Huang, L. Pareschi, and P. S¨ unnen. Consensus-based optimization on the sphere: convergence to global minimizers and machine learning. J. Mach. Learn. Res. , 22:Paper No. 237, 55, 2021

work page 2021

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Fornasier, H

M. Fornasier, H. Huang, L. Pareschi, and P. S¨ unnen. Anisotropic diffusion in consensus-based optimization on the sphere. SIAM J. Optim. , 32(3):1984–2012, 2022

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Fornasier, T

M. Fornasier, T. Klock, and K. Riedl. Consensus-based optimization methods converge globally. arXiv:2103.15130, 2021

work page arXiv 2021

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Fornasier, T

M. Fornasier, T. Klock, and K. Riedl. Convergence of anisotropic consensus-based opti- mization in mean-field law. In J. L. Jim´ enez Laredo, J. I. Hidalgo, and K. O. Babaagba, editors, Applications of Evolutionary Computation , pages 738–754, Cham, 2022. Springer International Publishing

work page 2022

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Grassi, H

S. Grassi, H. Huang, L. Pareschi, and J. Qiu. Mean-field particle swarm optimization. In Modeling and Simulation for Collective Dynamics , pages 127–193. World Scientific, 2023

work page 2023

[26] [26]

Grassi and L

S. Grassi and L. Pareschi. From particle swarm optimization to consensus based optimization: stochastic modeling and mean-field limit. Math. Models Methods Appl. Sci., 31(8):1625–1657, 2021. 25

work page 2021

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S.-Y. Ha, S. Jin, and D. Kim. Convergence of a first-order consensus-based global optimiza- tion algorithm. Math. Models Methods Appl. Sci. , 30(12):2417–2444, 2020

work page 2020

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S.-Y. Ha, S. Jin, and D. Kim. Convergence and error estimates for time-discrete consensus- based optimization algorithms. Numer. Math., 147(2):255–282, 2021

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S.-Y. Ha, M. Kang, and D. Kim. Emergent behaviors of high-dimensional Kuramoto models on Stiefel manifolds. Automatica, 136:Paper No. 110072, 2022

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Huang, J

H. Huang, J. Qiu, and K. Riedl. Consensus-based optimization for saddle point problems. arXiv:2212.12334, 2022

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Huang, J

H. Huang, J. Qiu, and K. Riedl. On the global convergence of particle swarm optimization methods. Applied Mathematics & Optimization , 88(2):30, 2023

work page 2023

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Kalise, A

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

work page 2023

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Kennedy and R

J. Kennedy and R. Eberhart. Particle swarm optimization. In Proceedings of ICNN’95 - International Conference on Neural Networks , volume 4, pages 1942–1948. IEEE, 1995

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J. Kim, M. Kang, D. Kim, S.-Y. Ha, and I. Yang. A stochastic consensus method for nonconvex optimization on the Stiefel manifold. In 2020 59th IEEE Conference on Decision and Control (CDC) , pages 1050–1057. IEEE, 2020

work page 2020

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Klamroth, M

K. Klamroth, M. Stiglmayr, and C. Totzeck. Consensus-based optimization for multi- objective problems: A multi-swarm approach. arXiv:2211.15737, 2022

work page arXiv 2022

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Ko, S.-Y

D. Ko, S.-Y. Ha, S. Jin, and D. Kim. Convergence analysis of the discrete consensus- based optimization algorithm with random batch interactions and heterogeneous noises. Mathematical Models and Methods in Applied Sciences , 32(06):1071–1107, 2022

work page 2022

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