Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency
Abstract
We prove uniform-in-time propagation of chaos and last-iterate consistency for a target-adapted Stein variational gradient descent (SVGD) flow on compact connected manifolds. The target has a smooth positive density, and the particles start independently from a fixed smooth nonnegative density ratio. The construction uses the Green--Bessel operator Qr,π=Aπ−1(Id+Aπ)−r of the reversible target Langevin generator. Sufficient Bessel smoothing gives a scalar kernel with finite diagonal, and a positive matrix lift realizes its potential force as a Stein velocity. Population and empirical flows then dissipate the same finite target discrepancy. Population entropy and the target spectral gap give decay of this discrepancy; a finite-time particle comparison reaches a time after which common-energy monotonicity controls every later time. The resulting expected uniform discrepancy is O((log N)−1/2), with a corresponding logarithmic W1 bound and consistency along every sequence tN→∞. We also prove an exact finite-mode approximation theorem with an explicit spatial-resolution error and a feature-factorized particle implementation. For confining Euclidean targets, we establish static kernel and moment results and give a conditional dynamical extension under explicit population-regularity and transport hypotheses.
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BibTeX
@misc{teolis2026targetadapted,
title = {{Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency}},
author = {Trevor Teolis and Maarten V. de Hoop},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08122},
eprint = {2609.08122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08122},
}
RIS
TY - UNPB
TI - Target-adapted Green-Bessel SVGD: uniform-in-time propagation of chaos and last-iterate consistency
AU - Trevor Teolis
AU - Maarten V. de Hoop
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08122
UR - https://arxiv.org/abs/2609.08122
AB - We prove uniform-in-time propagation of chaos and last-iterate consistency for a target-adapted Stein variational gradient descent (SVGD) flow on compact connected manifolds. The target has a smooth positive density, and the particles start independently from a fixed smooth nonnegative density ratio. The construction uses the Green--Bessel operator Q_r,π=A_π^−1(Id+A_π)^−r of the reversible target Langevin generator. Sufficient Bessel smoothing gives a scalar kernel with finite diagonal, and a positive matrix lift realizes its potential force as a Stein velocity. Population and empirical flows then dissipate the same finite target discrepancy. Population entropy and the target spectral gap give decay of this discrepancy; a finite-time particle comparison reaches a time after which common-energy monotonicity controls every later time. The resulting expected uniform discrepancy is O((log N)^−1/2), with a corresponding logarithmic W_1 bound and consistency along every sequence t_N→∞. We also prove an exact finite-mode approximation theorem with an explicit spatial-resolution error and a feature-factorized particle implementation. For confining Euclidean targets, we establish static kernel and moment results and give a conditional dynamical extension under explicit population-regularity and transport hypotheses.
ER -