Self-interacting CBO: Existence, uniqueness, and long-time convergence

Hicham Kouhkouh; Hui Huang

REVIEW 3 major objections 5 minor 20 references

Self-interacting CBO: Existence, uniqueness, and long-time convergence

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read One trajectory can replace the CBO particle swarm: a single self-interacting diffusion converges to the same invariant measure as the mean-field consensus-based optimization process, at a polynomial rate.

desk verdict A solid and genuinely new convergence theorem for a single-particle CBO is undermined by an over-sold abstract and a wrong supplementary 'global minimizer' step, but the core result deserves a referee. read the letter →

arxiv 2411.10295 v1 pith:JDJG3AEX submitted 2024-11-15 math.OC math.PR

Hui Huang , Hicham Kouhkouh This is my paper · ORCID

classification math.OCmath.PR MSC 60H1065C3537N4090C26

keywords self-interactingdiffusionconsensus-basedoptimizationinvariantmeasureglobalMcKean–Vlasovprocessmean-fieldlimitpersonalbestoccupation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces a self-interacting version of consensus-based optimization (CBO): one particle whose drift and diffusion depend on its own past occupation measure, not on a swarm of N interacting particles. The authors try to establish that this single-particle process and the mean-field CBO process share the same unique invariant measure, and that both converge to it polynomially fast. If true, running one trajectory replaces simulating an interacting particle system, and the consensus point built from the trajectory concentrates near a global minimizer as the parameter α grows. The global-minimization part is conditional on an unverified mass assumption on the invariant measure near the minimizer, which the paper states is still being worked on.

What carries the argument

The load-bearing object is the occupation measure $E_t[Y] = \frac{1}{t}\int_0^t \delta_{Y_r}\,dr$, used in place of the law $L_t[X]$ inside the consensus point $m_\alpha(\cdot) = \frac{\int x e^{-\alpha f(x)}\,\cdot(dx)}{\int e^{-\alpha f(x)}\,\cdot(dx)}$. The model also uses the rescaling $0<\kappa\ll1$ in the drift and the nondegenerate diffusion $\sigma(\alpha^{-1} I_d + D(Y_t-\kappa m_\alpha(E_t[Y])))$, which together rule out the Dirac invariant measures that plague the standard CBO dynamics. The convergence proof compares $Y$ and the mean-field $X$ with an auxiliary Markovian SDE in which $m_\alpha(\mu^*_\alpha)$ is frozen; a Gronwall-type estimate and weight classes $\Pi_1(\varepsilon),\Pi_2(\varepsilon)$ control the expected Wasserstein distance between weighted occupation measures.

What would settle it

For a chosen objective f, simulate the frozen Markovian process (18) long enough to approximate μ*_α, then evaluate $-\frac{1}{\alpha}\log\int e^{-\alpha f(x)}\,\mu^*_\alpha(dx)$ for increasing α. If this quantity does not approach f(x*) as α→∞, the mass hypothesis fails and the advertised global-minimization guarantee collapses.

Watch

Extended reading notes

Core claim

The central claim is that the rescaled mean-field CBO process (7) has a unique invariant measure μ*_α, and the single-particle self-interacting process (6) converges to it in the sense of expected squared Wasserstein-2 distance of its occupation measure, at a polynomial rate $t^{{-ε}}$. The proof couples both processes to an auxiliary Markovian process (18) with a frozen consensus point m_α(μ*_α), whose weighted occupation measure is shown to converge using comparison estimates for occupation measures. Because the same invariant measure is shared, the occupation measure of one trajectory can substitute for the N-particle empirical measure in the long-time limit. With an additional unverified Laplace-type mass condition, the consensus point built from the trajectory approximates a global minimizer as α→∞.

Load-bearing premise

The only unproved step connecting the invariant measure to the global minimum is that, as α grows, the invariant measure keeps a fixed amount of mass on the set where f is within ε of its best value; the paper says verifying this is ongoing work.

Editorial extensions

If this is right

The single-particle dynamics (6) can be used in place of the N-particle system (5) for long-time approximation of the CBO invariant measure, eliminating the need for N→∞.
Both the self-interacting and mean-field CBO processes converge to the same unique invariant measure at polynomial rate, strengthening the theoretical basis of CBO with Personal Best.
The rescaled mean-field CBO has a unique invariant measure even though it does not satisfy the usual dissipativity assumption, extending the class of McKean–Vlasov SDEs with uniqueness.
For large α, the consensus point built from the trajectory is expected to approximate $\kappa x_*$, so a single trajectory can approximately locate a global minimizer.
The multi-particle analogue inherits a rate combining particle number and time, giving a finite-time, finite-ensemble guarantee for the empirical occupation measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The unverified mass assumption is the only bridge from convergence to an invariant measure to convergence to a global minimizer; if it fails, the optimization guarantee weakens to convergence to some invariant measure, not necessarily concentrated near the minimum.
Since the method only uses moments through the consensus point, it may extend to nonsmooth objectives satisfying the growth and Lipschitz-type conditions; a natural test is to run the single-particle scheme on standard high-dimensional nonconvex benchmarks.
A practical implementation would need a finite-memory approximation of the occupation measure and a discretization scheme; the paper does not analyze the resulting discretization error, which is a natural next step.
The connection to CBO with Personal Best suggests that memory-based optimizers could be analyzed through the same occupation-measure comparison, potentially giving convergence rates for other trajectory-weighted consensus algorithms.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The real content of this paper is the polynomial convergence of the occupation measure of a single self-interacting particle to the unique invariant measure of a κ-rescaled mean-field CBO. That is new, and it is a nontrivial extension of the Du–Jiang–Li framework to a non-dissipative setting. The proof of Theorem 3.5 is structurally sound: Lemma 3.2 gives the needed one-sided inequality with explicit constants, and the rest is a careful application of [4] and [20]. Corollary 3.8, uniqueness of the invariant measure, follows cleanly. The paper is transparent about the regime λ > 8σ² and κ ≪ 1, and it does not hide the fact that the main step uses external machinery. I found nothing circular or fitted in the main argument.

The soft spot is exactly where the abstract overstates the result. The claim that the invariant measure 'approximates the global minimum' is not proven. The paper needs an α-uniform lower bound on μ*_α(A_ε), which the authors explicitly say is 'undergoing work.' Worse, the supplementary computation in §A.3, meant to close this gap, is mathematically wrong: the equality e^{-αf(x*)}/Z_α = ⟨η*_α, I_{x*}⟩ silently assumes an atom at x*, which an atomless μ*_α will not have. The Laplace limit only gives concentration in α-dependent neighborhoods, not point-mass convergence. This is not a minor typo; it is the only bridge from the invariant measure to global optimization, and it breaks. The connection to Personal Best rests on the same broken bridge.

That said, the broken bridge does not infect the main theorem. Theorem 3.5 and Corollary 3.8 stand on their own. The paper would be a solid contribution if the authors either prove the missing condition or simply drop the global-minimizer claim from the abstract and the formal part, presenting the result as convergence to a unique invariant measure of a rescaled CBO.

Who should read this? People working on consensus-based optimization and on self-interacting diffusions will find the convergence theorem and the uniqueness corollary useful and citable. The flaws are in the packaging, not in the core proof.

Recommendation: send this to a serious referee. A good referee will ask for the abstract to be corrected and the A.3 argument fixed or removed, but the substantive result merits the attention.

Referee Report

3 major / 5 minor

Summary. The paper introduces a single-particle self-interacting CBO dynamics (6) with an occupation-measure consensus point, together with generalized weighted versions (9)-(10). It claims that both the self-interacting process and the rescaled mean-field CBO (7) converge, in expected occupation-measure Wasserstein distance, to a common invariant measure μ*_α at a polynomial rate (Theorem 3.5), that μ*_α is unique (Corollary 3.8), and that μ*_α approximates a global minimizer through a Laplace-principle argument, thereby connecting the model to CBO with Personal Best. The convergence-to-invariant-measure part is supported by the estimates in Lemma 3.2 and external results [4,20]; the global-minimization bridge is not established.

Significance. If the convergence theorem is correct, the paper provides a useful single-particle alternative to N-particle CBO and extends the self-interaction framework of [4] to a consensus-based dynamics. The polynomial rate (17), the explicit weight classes, and the uniqueness result are concrete contributions. The paper is also honest in flagging the unverified α-uniform mass condition. However, the advertised application to global optimization is not established: the arguments after (16) and in Appendix A.3 contain a mathematical error, and the needed mass condition is left as future work. The present manuscript therefore establishes convergence to an α-dependent invariant measure, not a global-minimizer approximation; the significance is conditional on either a proof of the missing mass condition or a careful reframing of the claims.

major comments (3)

[Paragraph after Eq. (16) and Appendix A.3] The derivation that η*_α approximates δ_{x*} is mathematically invalid. From the Laplace limit lim_{α→∞} −(1/α) log Z_α = f(x*) one can conclude that e^{−f(x*)}/Z_α^{1/α} → 1, but raising this ratio to the power α is not justified, so the conclusion lim_{α→∞} e^{−αf(x*)}/Z_α = 1 does not follow. In addition, the equality e^{−αf(x*)}/Z_α = ⟨η*_α, I_{x*}⟩ silently drops the point mass μ*_α({x*}); for an atomless μ*_α the right-hand side is zero while the left-hand side is generally non-zero. The paper itself states that the required α-uniform lower bound μ*_α(A_ε) ≥ C_ε is 'undergoing work' (paragraph after (16)). Since the abstract's global-minimization claim rests entirely on this bridge, that advertised claim is not established.
[Abstract and Theorem 3.5] The abstract states that the dynamics 'converges to a unique invariant measure that approximates the global minimum'. Theorem 3.5 as proven only gives polynomial convergence of expected occupation measures E^ϑ_t[Y] and E^ϑ_t[X] to μ*_α in Wasserstein distance; it does not give convergence of the law of the process at fixed times, nor any information about the support of μ*_α. The statement E[X_∞] = κ m_α(μ*_α) in (15) is formal because X_∞ is not defined. The paper should either prove the global-minimizer statement under an explicit assumption or revise the abstract and Section 1 to advertise only convergence to the unique invariant measure.
[Appendix A.2, Proposition 3.4] The verification of Assumption (H1) of [20, Theorem 2.2] contains a constant mismatch. The display before the definition of ̃C_1 has coefficient 2σ²(1+κ) in the |x|² term and 2σ²(1+κ)κ C_1² in the ν(|·|²) term, but ̃C_1 is then defined with σ²(1+κ²) and ̃C_3 with σ²(1+κ)κ. Because ̃C_1 as defined is larger than the coefficient actually obtained, the inequality 2⟨b(x,ν),ν⟩+‖σ(x,ν)‖² ≤ −̃C_1|x|² + ̃C_2 + ̃C_3 ν(|·|²) does not follow as written. This is likely repairable by taking ̃C_1 = (2λ−λκ)−2σ²(1+κ) and ̃C_3 = λκC_1²+2σ²(1+κ)κC_1², but as it stands Proposition 3.4 relies on an unverified estimate.

minor comments (5)

[Assumption 2.1] Assumption 2.1 does not guarantee that the global minimizer x* exists; the paper should add an attainment condition (e.g., f coercive and continuous, or explicitly assume argmin f is nonempty) because x* appears in (16) and in the definition of A_ε.
[Theorem 3.11] In Theorem 3.11, the condition 'ϑ∈Π(ε2)' is almost certainly a typo for 'ϑ∈Π_2(ε2)'; please correct it to match the notation used in Theorem 3.5.
[Corollary 3.8] In Corollary 3.8, the application of Theorem 3.5 to a second invariant measure ̃μ*_α should state explicitly why ̃μ*_α ∈ P_{2,R} for the relevant R; this follows from the Lyapunov-type estimate but should be spelled out.
[Proof of Theorem 3.5, after Eq. (20)] The bound ∫_0^1 t^ε ∧ s^{-ε} ϑ_t(ds) ≤ ∫_0^1 t^{ε_2} ∧ s^{-ε_2} ϑ_t(ds) requires ε ≤ ε_2; since ε = ε_1 ∧ γε_2 ≤ ε_2 this is fine, but the monotonicity should be stated explicitly for the reader.
[Introduction, paragraph before Eq. (4)] The phrase 'as we shall later see, E[X_{t=∞}]≈κ x*' is informal because X_∞ is not a random variable; please rephrase using the invariant measure μ*_α, e.g., 'the mean of μ*_α is close to κ x*'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence-to-invariant-measure result is derived from external benchmarks [4], [10], [20], and the only self-cited lemma [14] is used in a formal, explicitly unfinished global-minimization argument rather than as a construction that reduces the conclusion to its inputs.

full rationale

The paper's main derivation chain is not circular. Theorem 3.5 and Corollary 3.8 are proved by checking dissipativity estimates (Lemma 3.2) and then importing weighted-occupation-measure convergence theorems from Du--Jiang--Li [4], with existence of the invariant measure from Zhang [20] and consensus-map estimates from Gerber--Hoffmann--Vaes [10]; none of these are fitted to the paper's conclusions and none cite the present authors' target result. The formal global-minimizer claim in Section 3 uses the self-cited [14, Lemma A.3] to pass from the assumed uniform lower bound mu*_alpha(A_eps) >= C_eps to the Laplace rate (16), and then infers that eta*_alpha approximates delta_{x*}. This step is not circular: it is an application of an external lemma under an assumption the authors explicitly say is 'undergoing work'. If anything, the inference in Appendix A.3 is mathematically invalid for atomless invariant measures (the displayed equality drops mu*_alpha({x*})), and the uniform mass hypothesis is unproved, so the advertised global-minimization conclusion is not established; but an invalid or incomplete inference is not an equivalence-by-construction, fitted-input, or self-citation chain. The central convergence statement has independent content and is benchmarked against external results. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new physical or mathematical entities are introduced. The self-interacting CBO is a new stochastic process and algorithm, not a new entity in the sense of a particle, force, or conserved quantity. The free parameters are the algorithm's design parameters κ, α, λ, σ, with the gap that the global-minimizer claim needs an unverified mass condition.

free parameters (3)

κ = 0 < κ < 1, sufficiently small (no quantitative threshold)
Introduced to break the point-mass invariant measures of standard CBO; Theorem 3.5 requires it small enough that a > 2b in Lemma 3.2.
α = any fixed α > 0 for the convergence theorem; unquantified 'large' for minimizer approximation
Inverse temperature in the weight exp(-α f); the global-minimizer claim is only argued in the limit α → ∞ and depends on an unverified mass condition.
λ, σ (regime) = λ > 8σ^2
Drift and diffusion coefficients; the dissipativity constants in Lemma 3.2 require this regime, which is assumed throughout the main theorems.

assumptions (6)

domain assumption Assumption 2.1 on f: bounded below, locally Lipschitz with |f(x)-f(y)| ≤ L_f(1+|x|+|y|)^s |x-y|, and growth c1(|x|^ℓ - 1) ≤ f - f ≤ c2(|x|^ℓ + 1).
Used in Lemma 3.1 for Lipschitz and growth of m_α and in the Laplace-principle heuristics; standard for CBO analysis.
standard math External results from [4] (Du-Jiang-Li): Proposition 3.1, Lemma 3.2, Lemma 4.1, Lemma 5.1 on self-interacting and mean-field processes.
The core convergence machinery is imported without proof; Theorem 3.5 and 3.11 rely directly on these lemmas.
standard math Lemma 3.1 estimates on m_α from [10, Corollary 3.3, Proposition A.3].
Provides local Lipschitz and growth bounds for the consensus map, used in Lemma 3.2.
standard math [20, Theorem 2.2] (Zhang) for existence of stationary distributions of distribution-dependent SDEs.
Applied in Proposition 3.4 to establish an invariant measure for (7) in P_{2,R}.
domain assumption Parameter conditions: λ > 8σ^2 and existence of κ ∈ (0,1) sufficiently small so that a > 2b and c > 0 in Lemma 3.2.
Needed for the contractive estimates; the paper argues such κ exists for fixed data but gives no quantitative bound.
ad hoc to paper Unverified mass condition: for every ε > 0 there is C_ε > 0 independent of α with μ*_α(A_ε) ≥ C_ε, A_ε = {x: e^{-f(x)} > e^{-f(x*)} - ε}.
Introduced after Eq. (16) to apply the Laplace principle and conclude the invariant measure concentrates on the global minimizer; labeled 'undergoing work'.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Self-interacting CBO: Existence, uniqueness, and long-time convergence." pith.science (2026). https://pith.science/paper/JDJG3AEX

@misc{pith2026241110295,
  author       = {Pith},
  title        = {Pith review of: Self-interacting CBO: Existence, uniqueness, and long-time convergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDJG3AEX}},
  note         = {Machine review of arXiv:2411.10295}
}

read the original abstract

A self-interacting dynamics that mimics the standard Consensus-Based Optimization (CBO) model is introduced. This single-particle dynamics is shown to converge to a unique invariant measure that approximates the global minimum of a given function. As an application, its connection to CBO with Personal Best introduced by C. Totzeck and M.-T. Wolfram (Math. Biosci. Eng., 2020) has been established.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 16 canonical work pages

[4]

K. Du, Y. Jiang, and J. Li, Empirical approximation to invariant measures for McKean– Vlasov processes: Mean-ﬁeld interaction vs self-interaction , Bernoulli 29 (2023), no. 3, 2492–2518

work page 2023
[20]

Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs , Electronic Journal of Probability 28 (2023), 1–34

S.-Q. Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs , Electronic Journal of Probability 28 (2023), 1–34. Appendix A. Supplementary material A.1. Proof of Lemma 3.2. We need to prove that ∃ a > 2b≥ 0 such that for all x, y∈ Rd and ( µ, ν)∈ Pp,R(Rd)× Pp(Rd) it holds that 2⟨b(x, µ)− b(y, ν), x− y⟩ +‖σ (x, µ)−...

work page 2023
[1]

A Carrillo, Y.-P

J. A Carrillo, Y.-P. Choi, C. Totzeck, and O. Tse, An analytical framework for consensus-based global optimization method , Mathematical Models and Methods in Applied Sciences 28 (2018), no. 06, 1037– 1066

work page 2018
[2]

A Carrillo, S

J. A Carrillo, S. Jin, L. Li, and Y. Zhu, A consensus-based global optimization method for high dime n- sional machine learning problems , ESAIM: Control, Optimisation and Calculus of Variations 27 (2021), S5

work page 2021
[3]

Dembo and O

A. Dembo and O. Zeitouni, Large deviations techniques and applications , Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010

work page 2010
[5]

K. Du, Z. Ren, F. Suciu, and S. Wang, Self-interacting approximation to McKean-Vlasov long-ti me limit: a Markov chain Monte Carlo method , preprint arXiv:2311.11428 (2023)

arXiv 2023
[6]

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 (2020), no. 14, 2725–2751

work page 2020
[7]

3, 1984–2012

, Anisotropic diﬀusion in consensus-based optimization on t he sphere , SIAM Journal on Opti- mization 32 (2022), no. 3, 1984–2012

work page 2022

Show all 20 references

[8]

Fornasier, T

M. Fornasier, T. Klock, and K. Riedl, Consensus-based optimization methods converge globally , SIAM Journal on Optimization 34 (2024), no. 3, 2973–3004

2024
[9]

Fornasier, L

M. Fornasier, L. Pareschi, H. Huang, and P. S¨ unnen, Consensus-based optimization on the sphere: Convergence to global minimizers and machine learning , Journal of Machine Learning Research 22 (2021), no. 237, 1–55

2021
[10]

N. J. Gerber, F. Hoﬀmann, and U. Vaes, Mean-ﬁeld limits for consensus-based optimization and sam - pling, arXiv preprint arXiv:2312.07373 (2023)

2023
[11]

Herty, Y

M. Herty, Y. Huang, D. Kalise, and H. Kouhkouh, A multiscale Consensus-Based algorithm for multi- level optimization , preprint arXiv:2407.09257 (2024)

2024 arXiv
[12]

Huang and H

H. Huang and H. Kouhkouh, Uniform-in-time mean-ﬁeld limit estimate for the Consensu s-Based Opti- mization, preprint arXiv:2411.03986 (2024)

2024 arXiv
[13]

Huang and J

H. Huang and J. Qiu, On the mean-ﬁeld limit for the consensus-based optimizatio n, Mathematical Methods in the Applied Sciences 45 (2022), no. 12, 7814–7831

2022
[14]

Huang, J

H. Huang, J. Qiu, and K. Riedl, Consensus-based optimization for saddle point problems , SIAM Journal on Control and Optimization 62 (2024), no. 2, 1093–1121

2024
[15]

Kouhkouh, A one-player approach to ergodic Mean-Field Games and Mean- Field Control , (submit- ted)

H. Kouhkouh, A one-player approach to ergodic Mean-Field Games and Mean- Field Control , (submit- ted)
[16]

P. D. Miller, Applied asymptotic analysis , Vol. 75, American Mathematical Soc., 2006. 10 HUI HUANG AND HICHAM KOUHKOUH

2006
[17]

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 (2017), no. 01, 183–204

2017
[18]

Totzeck and M.-T

C. Totzeck and M.-T. Wolfram, Consensus-based global optimization with personal best , Mathematical Biosciences and Engineering 17 (2020), no. 5, 6026–6044

2020
[19]

Wang, Distribution dependent SDEs for Landau type equations , Stochastic Processes and their Applications 128 (2018), no

F.-Y. Wang, Distribution dependent SDEs for Landau type equations , Stochastic Processes and their Applications 128 (2018), no. 2, 595–621

2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

K. Du, Y. Jiang, and J. Li, Empirical approximation to invariant measures for McKean– Vlasov processes: Mean-ﬁeld interaction vs self-interaction , Bernoulli 29 (2023), no. 3, 2492–2518

work page 2023

[2] [20]

Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs , Electronic Journal of Probability 28 (2023), 1–34

S.-Q. Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs , Electronic Journal of Probability 28 (2023), 1–34. Appendix A. Supplementary material A.1. Proof of Lemma 3.2. We need to prove that ∃ a > 2b≥ 0 such that for all x, y∈ Rd and ( µ, ν)∈ Pp,R(Rd)× Pp(Rd) it holds that 2⟨b(x, µ)− b(y, ν), x− y⟩ +‖σ (x, µ)−...

work page 2023

[3] [1]

A Carrillo, Y.-P

J. A Carrillo, Y.-P. Choi, C. Totzeck, and O. Tse, An analytical framework for consensus-based global optimization method , Mathematical Models and Methods in Applied Sciences 28 (2018), no. 06, 1037– 1066

work page 2018

[4] [2]

A Carrillo, S

J. A Carrillo, S. Jin, L. Li, and Y. Zhu, A consensus-based global optimization method for high dime n- sional machine learning problems , ESAIM: Control, Optimisation and Calculus of Variations 27 (2021), S5

work page 2021

[5] [3]

Dembo and O

A. Dembo and O. Zeitouni, Large deviations techniques and applications , Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010

work page 2010

[6] [5]

K. Du, Z. Ren, F. Suciu, and S. Wang, Self-interacting approximation to McKean-Vlasov long-ti me limit: a Markov chain Monte Carlo method , preprint arXiv:2311.11428 (2023)

arXiv 2023

[7] [6]

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 (2020), no. 14, 2725–2751

work page 2020

[8] [7]

3, 1984–2012

, Anisotropic diﬀusion in consensus-based optimization on t he sphere , SIAM Journal on Opti- mization 32 (2022), no. 3, 1984–2012

work page 2022