Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915998899109888 |
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| author | Chizat, Lénaïc Colombo, Maria Colombo, Roberto Fernández-Real, Xavier |
| author_facet | Chizat, Lénaïc Colombo, Maria Colombo, Roberto Fernández-Real, Xavier |
| contents | Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09456 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow Chizat, Lénaïc Colombo, Maria Colombo, Roberto Fernández-Real, Xavier Machine Learning Analysis of PDEs Optimization and Control 35Q68, 35Q62, 35B40, 35Q84 Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies. |
| title | Quantitative Local Convergence of Mean-Field Stein Variational Gradient Flow |
| topic | Machine Learning Analysis of PDEs Optimization and Control 35Q68, 35Q62, 35B40, 35Q84 |
| url | https://arxiv.org/abs/2605.09456 |