Stein Variational Gradient Descent dynamics for highly concentrated kernels

Jakub Skrzeczkowski; Jethro Warnett; Jos\'e A. Carrillo

arxiv: 2605.03627 · v1 · submitted 2026-05-05 · 🧮 math.AP

Stein Variational Gradient Descent dynamics for highly concentrated kernels

Jos\'e A. Carrillo , Jakub Skrzeczkowski , Jethro Warnett This is my paper

Pith reviewed 2026-05-07 15:10 UTC · model grok-4.3

classification 🧮 math.AP

keywords Stein Variational Gradient DescentSVGDsingular limitWasserstein gradient flowquadratic mobilitykernel bandwidthStein-log-Sobolev inequalitiesnonlocal to local convergence

0 comments

The pith

As the kernel bandwidth in SVGD tends to zero, the nonlocal particle dynamics converge to a local evolution equation that is a Wasserstein gradient flow with quadratic mobility.

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

The paper studies the singular limit of Stein Variational Gradient Descent when the kernel bandwidth shrinks to zero. In this regime the nonlocal interacting-particle system collapses to a local PDE that can be viewed as a Wasserstein gradient flow whose mobility is quadratic in the density. The result is proved first for integrable kernels and then for weighted kernels, where Stein-log-Sobolev inequalities supply the required functional estimates. The analysis clarifies how a practical sampling algorithm transitions from nonlocal to local dynamics under kernel concentration.

Core claim

In the singular limit where the kernel bandwidth tends to zero, the nonlocal SVGD dynamics converge to a local evolution equation that can be formally interpreted as a Wasserstein gradient flow with quadratic mobility. The convergence holds in two settings: for integrable kernels and, with the aid of Stein-log-Sobolev inequalities, for weighted kernels.

What carries the argument

The singular limit of vanishing kernel bandwidth, which turns the nonlocal SVGD particle system into a local Wasserstein gradient flow with quadratic mobility.

If this is right

SVGD particle updates become indistinguishable from a local continuum gradient flow once the kernel is sufficiently concentrated.
The quadratic mobility term governs the speed of the limiting density evolution.
Functional control in the weighted case rests on Stein-log-Sobolev inequalities rather than standard Sobolev embeddings.
The limit procedure supplies a rigorous justification for using SVGD as an approximation to local mean-field dynamics.

Where Pith is reading between the lines

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

The local limit equation could be discretized directly to produce new deterministic sampling schemes that avoid kernel summation.
Similar concentration arguments may apply to other nonlocal particle methods whose kernels admit Stein-type functional inequalities.
The quadratic mobility suggests that the limiting flow may preserve certain convexity or contractivity properties inherited from the Wasserstein geometry.

Load-bearing premise

The kernel bandwidth must tend to zero while Stein-log-Sobolev inequalities remain available to control the weighted-kernel case.

What would settle it

Direct numerical comparison of SVGD particle trajectories at successively smaller bandwidths against the solution of the candidate local PDE on the same initial data; systematic deviation would falsify the claimed convergence.

read the original abstract

Stein Variational Gradient Descent (SVGD) is a widely used in practice algorithm for scalable sampling with deterministic particle updates. We study its behavior in the singular limit where the kernel bandwidth tends to zero. In this regime, we show that the nonlocal SVGD dynamics converge to a local evolution equation that can be formally interpreted as a Wasserstein gradient flow with quadratic mobility. We analyze this singular limit in two settings: integrable kernels and weighted kernels. In the weighted case, the proof is supported by recently established Stein-log-Sobolev inequalities, which provide the necessary functional control. Overall, our results clarify how SVGD collapses from a nonlocal interacting particle system to a local gradient-flow dynamics as the kernel concentrates.

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

SVGD with concentrating kernels converges to a local quadratic-mobility Wasserstein flow in both kernel regimes, but the weighted case hinges on whether Stein-log-Sobolev constants stay uniform as bandwidth vanishes.

read the letter

The main result is a singular-limit statement: as the kernel bandwidth goes to zero, the nonlocal SVGD particle dynamics converge to a local PDE that formally matches a Wasserstein gradient flow with quadratic mobility. The authors treat integrable kernels and weighted kernels separately, invoking Stein-log-Sobolev inequalities for the functional estimates needed in the weighted setting to pass to the limit and identify the equation. This organizes the transition from nonlocal to local behavior in a direct way that prior SVGD literature had not spelled out for this concentration regime. The interpretation as a gradient flow is a useful organizing principle for understanding the algorithm's continuum limit. The argument relies on standard singular-limit techniques plus the external inequalities rather than any internal fitting, so there is no obvious circularity. The math is formally grounded on the abstract level, with no invented entities or free parameters flagged. The soft spot is exactly the one raised in the stress test: the paper does not appear to supply explicit uniformity of the Stein-log-Sobolev constants with respect to the bandwidth parameter h. If those constants deteriorate as h tends to zero, the a-priori bounds required for compactness or limit identification could fail even though the inequalities hold for each fixed h. Without seeing the detailed error estimates or boundary-term handling, it is hard to confirm the constants remain controlled. This paper is aimed at analysts working on interacting particle systems and deterministic sampling methods. Readers who want a rigorous account of how SVGD collapses to local dynamics will find it relevant and worth reading. It deserves a serious referee because the claim is concrete, the setting is practically motivated, and the approach is standard but applied to a new limit. I would send it to peer review and ask the referees to check the uniformity of the functional inequalities in the weighted case.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the singular limit of Stein Variational Gradient Descent (SVGD) dynamics as the kernel bandwidth tends to zero. It shows that the nonlocal SVGD dynamics converge to a local evolution equation that can be formally interpreted as a Wasserstein gradient flow with quadratic mobility. The analysis is carried out separately for integrable kernels and weighted kernels, with the weighted case relying on Stein-log-Sobolev inequalities for the necessary functional control.

Significance. If the convergence result is established rigorously, the paper would provide a significant contribution by clarifying the connection between nonlocal particle systems in SVGD and local gradient flow dynamics. This has potential implications for the theoretical foundations of sampling algorithms. The use of recently established Stein-log-Sobolev inequalities is a positive aspect, demonstrating engagement with current research in functional inequalities. However, the result's validity hinges on the uniformity of the constants in these inequalities with respect to the bandwidth parameter.

major comments (2)

In the weighted-kernel analysis, the Stein-log-Sobolev inequalities are used to obtain the functional control needed to pass to the limit as h→0. The manuscript does not establish (or cite) any bound showing that the constants remain uniform or mildly dependent on h; deterioration of these constants would invalidate the uniform a-priori estimates required for compactness and limit identification. This is the load-bearing step for the main result in the weighted setting.
The local limit equation is described as 'formally interpreted' as a Wasserstein gradient flow with quadratic mobility. The manuscript should supply a precise derivation or identification of the energy and mobility in the limit (e.g., via the continuity equation and the quadratic structure), rather than leaving the interpretation at the formal level, since this is part of the central claim.

minor comments (2)

The introduction would benefit from an explicit statement of the two main theorems, including the precise form of the local evolution equation obtained in each case.
Notation for the rescaled kernel K_h and the bandwidth parameter h should be introduced once and used consistently; occasional shifts between h and other concentration parameters are distracting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable feedback on our work concerning the singular limit of SVGD dynamics. The comments highlight important aspects that will improve the rigor and clarity of the manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

Referee: In the weighted-kernel analysis, the Stein-log-Sobolev inequalities are used to obtain the functional control needed to pass to the limit as h→0. The manuscript does not establish (or cite) any bound showing that the constants remain uniform or mildly dependent on h; deterioration of these constants would invalidate the uniform a-priori estimates required for compactness and limit identification. This is the load-bearing step for the main result in the weighted setting.

Authors: We thank the referee for pointing out this critical aspect. The uniformity of the constants in the Stein-log-Sobolev inequalities with respect to the bandwidth h is indeed essential for obtaining uniform estimates. While the cited references establish the inequalities, they do not explicitly address the dependence on h. In the revised manuscript, we will include an appendix or subsection that derives the h-uniformity of the constants for the weighted kernels under consideration. This will involve a careful tracking of the constants through the proof, demonstrating that they remain bounded independently of h as h approaches zero. This addition will ensure the compactness arguments hold uniformly. revision: yes
Referee: The local limit equation is described as 'formally interpreted' as a Wasserstein gradient flow with quadratic mobility. The manuscript should supply a precise derivation or identification of the energy and mobility in the limit (e.g., via the continuity equation and the quadratic structure), rather than leaving the interpretation at the formal level, since this is part of the central claim.

Authors: We agree that elevating the interpretation from formal to precise would strengthen the central claim. In the revision, we will add a detailed derivation in Section 3 or a new subsection. Starting from the weak form of the SVGD continuity equation, we will pass to the limit as h→0 and identify the limiting velocity field. This will reveal the energy functional as the Kullback-Leibler divergence or appropriate relative entropy, and the mobility as quadratic in the density gradient, consistent with the Wasserstein structure. We will explicitly compute the first variation and show the quadratic mobility term arises from the concentrating kernel. The abstract and introduction will be updated accordingly to reflect this precise identification. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to external Stein-log-Sobolev inequalities; derivation remains independent

full rationale

The paper establishes convergence of nonlocal SVGD particle dynamics to a local Wasserstein gradient flow with quadratic mobility in the h→0 singular limit, separately for integrable and weighted kernels. For the weighted case it invokes recently established Stein-log-Sobolev inequalities solely to supply a priori functional control and compactness, then applies standard singular-limit passage techniques. These inequalities are treated as external input rather than derived or presupposed inside the present work; the target convergence statement is not equivalent to them by construction, nor is any parameter fitted to data and relabeled as prediction. No self-definitional, fitted-input, or ansatz-smuggling reductions appear. The single minor citation therefore raises the score only to 2 while leaving the central claim with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of Stein-log-Sobolev inequalities for the weighted kernels and on standard compactness or stability arguments for singular limits of nonlocal PDEs; no free parameters or new entities are introduced.

axioms (1)

domain assumption Stein-log-Sobolev inequalities hold and provide the necessary functional control for weighted kernels
Explicitly invoked to support the proof in the weighted-kernel setting.

pith-pipeline@v0.9.0 · 5417 in / 1125 out tokens · 39112 ms · 2026-05-07T15:10:29.381244+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 1 internal anchor

[1]

Ambrosio, N

L. Ambrosio, N. Fusco, and D. Pallara.Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

work page 2000
[2]

Aronszajn and K

N. Aronszajn and K. T. Smith. Theory of Bessel potentials. I.Ann. Inst. Fourier (Grenoble), 11:385–475, 1961

work page 1961
[3]

Banerjee, K

S. Banerjee, K. Balasubramanian, and P. Ghosal. Improved finite-particle convergence rates for Stein variational gradient descent.arXiv preprint arXiv:2409.08469, 2025

work page arXiv 2025
[4]

J. A. Carrillo, K. Craig, and F. S. Patacchini. A blob method for diffusion.Calc. Var. Partial Differential Equations, 58(2):Paper No. 53, 53, 2019

work page 2019
[5]

J. A. Carrillo, A. Esposito, J. Skrzeczkowski, and J. S.-H. Wu. Nonlocal particle approximation for linear and fast diffusion equations.arXiv preprint arXiv:2408.02345, 2024

work page arXiv 2024
[6]

J. A. Carrillo, D. G´ omez-Castro, and J. L. V´ azquez. A fast regularisation of a Newtonian vortex equation.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 39(3):705–747, 2022

work page 2022
[7]

J. A. Carrillo, D. G´ omez-Castro, and J. L. V´ azquez. Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility.Adv. Nonlinear Anal., 11(1):937–967, 2022

work page 2022
[8]

J. A. Carrillo, S. Lisini, G. Savar´ e, and D. Slepˇ cev. Nonlinear mobility continuity equations and gener- alized displacement convexity.J. Funct. Anal., 258(4):1273–1309, 2010

work page 2010
[9]

J. A. Carrillo and J. Skrzeczkowski. Convergence and stability results for the particle system in the Stein gradient descent method.Math. Comp., 94(354):1793–1814, 2025

work page 2025
[10]

J. A. Carrillo, J. Skrzeczkowski, and J. Warnett. The Stein-log-Sobolev inequality and the exponen- tial rate of convergence for the continuous Stein variational gradient descent method.arXiv preprint arXiv:2412.10295, 2024

work page arXiv 2024
[11]

Chewi, T

S. Chewi, T. Le Gouic, C. Lu, T. Maunu, and P. Rigollet. SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence. InAdvances in Neural Information Processing Systems, volume 33, pages 2098–2109. Curran Associates, Inc., 2020. 48 JOS ´E A. CARRILLO, JAKUB SKRZECZKOWSKI, AND JETHRO WARNETT

work page 2098
[12]

Constantin, W

P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation.Comm. Math. Phys., 165(1):207–209, 1994

work page 1994
[13]

Craig, M

K. Craig, M. Jacobs, and O. Turanova. Nonlocal approximation of slow and fast diffusion.J. Differential Equations, 426:782–852, 2025

work page 2025
[14]

Davoli, L

E. Davoli, L. Scarpa, and L. Trussardi. Nonlocal-to-local convergence of Cahn-Hilliard equations: Neu- mann boundary conditions and viscosity terms.Arch. Ration. Mech. Anal., 239(1):117–149, 2021

work page 2021
[15]

A. C. de Courcel and C. Elbar. On a repulsion model with Coulomb interaction and nonlinear mobility. arXiv preprint arXiv:2510.16894, 2025

work page arXiv 2025
[16]

R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3):511–547, 1989

work page 1989
[17]

Duncan, N

A. Duncan, N. N¨ usken, and L. Szpruch. On the geometry of Stein variational gradient descent.J. Mach. Learn. Res., 24:Paper No. [56], 39, 2023

work page 2023
[18]

Elbar, B

C. Elbar, B. Perthame, and J. Skrzeczkowski. On the limit problem arising in the kinetic derivation of a Cahn-Hilliard equation.Comm. Math. Phys., 405(11):Paper No. 273, 16, 2024

work page 2024
[19]

Elbar and J

C. Elbar and J. Skrzeczkowski. Degenerate Cahn-Hilliard equation: from nonlocal to local.J. Differen- tial Equations, 364:576–611, 2023

work page 2023
[20]

Y. Feng, D. Wang, and Q. Liu. Learning to draw samples with amortized Stein variational gradient descent. InProceedings of the Conference on Uncertainty in Artificial Intelligence (UAI), 2017

work page 2017
[21]

Haarnoja, H

T. Haarnoja, H. Tang, P. Abbeel, and S. Levine. Reinforcement learning with deep energy-based policies. In D. Precup and Y. W. Teh, editors,Proceedings of the 34th International Conference on Machine Learning, volume 70 ofProceedings of Machine Learning Research, pages 1352–1361. PMLR, 06–11 Aug 2017

work page 2017
[22]

P. R. Halmos and V. S. Sunder.Bounded integral operators onL 2 spaces. Springer Science & Business Media, 2012

work page 2012
[23]

Y. He, K. Balasubramanian, S. Banerjee, and P. Ghosal. Finite-particle rates for regularized Stein variational gradient descent.arXiv preprint arXiv:2602.05172, 2026

work page internal anchor Pith review arXiv 2026
[24]

Y. He, K. Balasubramanian, B. K. Sriperumbudur, and J. Lu. Regularized Stein variational gradient flow.Found. Comput. Math., 25(4):1199–1257, 2025

work page 2025
[25]

Korba, A

A. Korba, A. Salim, M. Arbel, G. Luise, and A. Gretton. A non-asymptotic analysis for Stein variational gradient descent. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, editors,Advances in Neural Information Processing Systems, volume 33, pages 4672–4682. Curran Associates, Inc., 2020

work page 2020
[26]

Lambert, F

A. Lambert, F. Ramos, B. Boots, D. Fox, and A. Fishman. Stein variational model predictive control. In J. Kober, F. Ramos, and C. Tomlin, editors,Proceedings of the 2020 Conference on Robot Learning, volume 155 ofProceedings of Machine Learning Research, pages 1278–1297. PMLR, 16–18 Nov 2021. SVGD DYNAMICS FOR HIGHLY CONCENTRATED KERNELS 49

work page 2020
[27]

Q. Liu. Stein variational gradient descent as gradient flow.Advances in Neural Information Processing Systems, 30, 2017

work page 2017
[28]

Liu and D

Q. Liu and D. Wang. Stein variational gradient descent: a general purpose Bayesian inference algorithm. Proc. 30th Int. Conf. Neural Inf. Proc. Syst., page 2378–2386, 2016

work page 2016
[29]

X. Liu, X. Tong, and Q. Liu. Sampling with trusthworthy constraints: A variational gradient framework. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Information Processing Systems, volume 34, pages 23557–23568. Curran Associates, Inc., 2021

work page 2021
[30]

J. Lu, Y. Lu, and J. Nolen. Scaling limit of the Stein variational gradient descent: the mean field regime. SIAM J. Math. Anal., 51(2):648–671, 2019

work page 2019
[31]

Salim, L

A. Salim, L. Sun, and P. Richtarik. A convergence theory for SVGD in the population limit under Talagrand’s inequality T1. InProceedings of the 39th International Conference on Machine Learning, volume 162 ofProceedings of Machine Learning Research, pages 19139–19152. PMLR, 17–23 Jul 2022

work page 2022
[32]

L. Sun, A. Karagulyan, and P. Richtarik. Convergence of Stein variational gradient descent under a weaker smoothness condition. InProceedings of The 26th International Conference on Artificial In- telligence and Statistics, volume 206 ofProceedings of Machine Learning Research, pages 3693–3717. PMLR, 25–27 Apr 2023

work page 2023
[33]

L. Xu, A. Korba, and D. Slepˇ cev. Accurate quantization of measures via interacting particle-based optimization. In K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, editors, Proceedings of the 39th International Conference on Machine Learning, volume 162 ofProceedings of Machine Learning Research, pages 24576–24595. PMLR, 17–23 Ju...

work page 2022

[1] [1]

Ambrosio, N

L. Ambrosio, N. Fusco, and D. Pallara.Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

work page 2000

[2] [2]

Aronszajn and K

N. Aronszajn and K. T. Smith. Theory of Bessel potentials. I.Ann. Inst. Fourier (Grenoble), 11:385–475, 1961

work page 1961

[3] [3]

Banerjee, K

S. Banerjee, K. Balasubramanian, and P. Ghosal. Improved finite-particle convergence rates for Stein variational gradient descent.arXiv preprint arXiv:2409.08469, 2025

work page arXiv 2025

[4] [4]

J. A. Carrillo, K. Craig, and F. S. Patacchini. A blob method for diffusion.Calc. Var. Partial Differential Equations, 58(2):Paper No. 53, 53, 2019

work page 2019

[5] [5]

J. A. Carrillo, A. Esposito, J. Skrzeczkowski, and J. S.-H. Wu. Nonlocal particle approximation for linear and fast diffusion equations.arXiv preprint arXiv:2408.02345, 2024

work page arXiv 2024

[6] [6]

J. A. Carrillo, D. G´ omez-Castro, and J. L. V´ azquez. A fast regularisation of a Newtonian vortex equation.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 39(3):705–747, 2022

work page 2022

[7] [7]

J. A. Carrillo, D. G´ omez-Castro, and J. L. V´ azquez. Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility.Adv. Nonlinear Anal., 11(1):937–967, 2022

work page 2022

[8] [8]

J. A. Carrillo, S. Lisini, G. Savar´ e, and D. Slepˇ cev. Nonlinear mobility continuity equations and gener- alized displacement convexity.J. Funct. Anal., 258(4):1273–1309, 2010

work page 2010

[9] [9]

J. A. Carrillo and J. Skrzeczkowski. Convergence and stability results for the particle system in the Stein gradient descent method.Math. Comp., 94(354):1793–1814, 2025

work page 2025

[10] [10]

J. A. Carrillo, J. Skrzeczkowski, and J. Warnett. The Stein-log-Sobolev inequality and the exponen- tial rate of convergence for the continuous Stein variational gradient descent method.arXiv preprint arXiv:2412.10295, 2024

work page arXiv 2024

[11] [11]

Chewi, T

S. Chewi, T. Le Gouic, C. Lu, T. Maunu, and P. Rigollet. SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence. InAdvances in Neural Information Processing Systems, volume 33, pages 2098–2109. Curran Associates, Inc., 2020. 48 JOS ´E A. CARRILLO, JAKUB SKRZECZKOWSKI, AND JETHRO WARNETT

work page 2098

[12] [12]

Constantin, W

P. Constantin, W. E, and E. S. Titi. Onsager’s conjecture on the energy conservation for solutions of Euler’s equation.Comm. Math. Phys., 165(1):207–209, 1994

work page 1994

[13] [13]

Craig, M

K. Craig, M. Jacobs, and O. Turanova. Nonlocal approximation of slow and fast diffusion.J. Differential Equations, 426:782–852, 2025

work page 2025

[14] [14]

Davoli, L

E. Davoli, L. Scarpa, and L. Trussardi. Nonlocal-to-local convergence of Cahn-Hilliard equations: Neu- mann boundary conditions and viscosity terms.Arch. Ration. Mech. Anal., 239(1):117–149, 2021

work page 2021

[15] [15]

A. C. de Courcel and C. Elbar. On a repulsion model with Coulomb interaction and nonlinear mobility. arXiv preprint arXiv:2510.16894, 2025

work page arXiv 2025

[16] [16]

R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98(3):511–547, 1989

work page 1989

[17] [17]

Duncan, N

A. Duncan, N. N¨ usken, and L. Szpruch. On the geometry of Stein variational gradient descent.J. Mach. Learn. Res., 24:Paper No. [56], 39, 2023

work page 2023

[18] [18]

Elbar, B

C. Elbar, B. Perthame, and J. Skrzeczkowski. On the limit problem arising in the kinetic derivation of a Cahn-Hilliard equation.Comm. Math. Phys., 405(11):Paper No. 273, 16, 2024

work page 2024

[19] [19]

Elbar and J

C. Elbar and J. Skrzeczkowski. Degenerate Cahn-Hilliard equation: from nonlocal to local.J. Differen- tial Equations, 364:576–611, 2023

work page 2023

[20] [20]

Y. Feng, D. Wang, and Q. Liu. Learning to draw samples with amortized Stein variational gradient descent. InProceedings of the Conference on Uncertainty in Artificial Intelligence (UAI), 2017

work page 2017

[21] [21]

Haarnoja, H

T. Haarnoja, H. Tang, P. Abbeel, and S. Levine. Reinforcement learning with deep energy-based policies. In D. Precup and Y. W. Teh, editors,Proceedings of the 34th International Conference on Machine Learning, volume 70 ofProceedings of Machine Learning Research, pages 1352–1361. PMLR, 06–11 Aug 2017

work page 2017

[22] [22]

P. R. Halmos and V. S. Sunder.Bounded integral operators onL 2 spaces. Springer Science & Business Media, 2012

work page 2012

[23] [23]

Y. He, K. Balasubramanian, S. Banerjee, and P. Ghosal. Finite-particle rates for regularized Stein variational gradient descent.arXiv preprint arXiv:2602.05172, 2026

work page internal anchor Pith review arXiv 2026

[24] [24]

Y. He, K. Balasubramanian, B. K. Sriperumbudur, and J. Lu. Regularized Stein variational gradient flow.Found. Comput. Math., 25(4):1199–1257, 2025

work page 2025

[25] [25]

Korba, A

A. Korba, A. Salim, M. Arbel, G. Luise, and A. Gretton. A non-asymptotic analysis for Stein variational gradient descent. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, editors,Advances in Neural Information Processing Systems, volume 33, pages 4672–4682. Curran Associates, Inc., 2020

work page 2020

[26] [26]

Lambert, F

A. Lambert, F. Ramos, B. Boots, D. Fox, and A. Fishman. Stein variational model predictive control. In J. Kober, F. Ramos, and C. Tomlin, editors,Proceedings of the 2020 Conference on Robot Learning, volume 155 ofProceedings of Machine Learning Research, pages 1278–1297. PMLR, 16–18 Nov 2021. SVGD DYNAMICS FOR HIGHLY CONCENTRATED KERNELS 49

work page 2020

[27] [27]

Q. Liu. Stein variational gradient descent as gradient flow.Advances in Neural Information Processing Systems, 30, 2017

work page 2017

[28] [28]

Liu and D

Q. Liu and D. Wang. Stein variational gradient descent: a general purpose Bayesian inference algorithm. Proc. 30th Int. Conf. Neural Inf. Proc. Syst., page 2378–2386, 2016

work page 2016

[29] [29]

X. Liu, X. Tong, and Q. Liu. Sampling with trusthworthy constraints: A variational gradient framework. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Information Processing Systems, volume 34, pages 23557–23568. Curran Associates, Inc., 2021

work page 2021

[30] [30]

J. Lu, Y. Lu, and J. Nolen. Scaling limit of the Stein variational gradient descent: the mean field regime. SIAM J. Math. Anal., 51(2):648–671, 2019

work page 2019

[31] [31]

Salim, L

A. Salim, L. Sun, and P. Richtarik. A convergence theory for SVGD in the population limit under Talagrand’s inequality T1. InProceedings of the 39th International Conference on Machine Learning, volume 162 ofProceedings of Machine Learning Research, pages 19139–19152. PMLR, 17–23 Jul 2022

work page 2022

[32] [32]

L. Sun, A. Karagulyan, and P. Richtarik. Convergence of Stein variational gradient descent under a weaker smoothness condition. InProceedings of The 26th International Conference on Artificial In- telligence and Statistics, volume 206 ofProceedings of Machine Learning Research, pages 3693–3717. PMLR, 25–27 Apr 2023

work page 2023

[33] [33]

L. Xu, A. Korba, and D. Slepˇ cev. Accurate quantization of measures via interacting particle-based optimization. In K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato, editors, Proceedings of the 39th International Conference on Machine Learning, volume 162 ofProceedings of Machine Learning Research, pages 24576–24595. PMLR, 17–23 Ju...

work page 2022