Riesz-Kernel Stein Variational Gradient Descent: Renormalized Entropy and Long-Time Particle Limits

We provide finite-particle convergence rates for the Stein Variational Gradient Descent (SVGD) algorithm in the Kernelized Stein Discrepancy ($\mathsf{KSD}$) and Wasserstein-2 metrics. Our key insight is that the time derivative of the relative entropy between the joint density of $N$ particle locations and the $N$-fold product target measure, starting from a regular initial distribution, splits into a dominant negative part' proportional to $N$ times the expected $\mathsf{KSD}^2$ and a smaller positive part'. This observation leads to $\mathsf{KSD}$ rates of order $1/\sqrt{N}$, in both continuous and discrete time, providing a near optimal (in the sense of matching the corresponding i.i.d. rates) double exponential improvement over the recent result by Shi and Mackey (2024). Under mild assumptions on the kernel and potential, these bounds also grow polynomially in the dimension $d$. By adding a bilinear component to the kernel, the above approach is used to further obtain Wasserstein-2 convergence in continuous time. For the case of `bilinear + Matérn' kernels, we derive Wasserstein-2 rates that exhibit a curse-of-dimensionality similar to the i.i.d. setting. We also obtain marginal convergence and long-time propagation of chaos results for the time-averaged particle laws.

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.

Particle-based variational inference (ParVI) methods approximate an intractable target distribution by evolving an ensemble of interacting samples. Existing approaches rely predominantly on kernel-based repulsion (e.g., SVGD), which suffers from variance collapse in high dimensions and mode collapse on multimodal targets -- pathologies caused by the absence of global transport structure. We introduce entropic transport descent (ETD), a ParVI family that frames each particle update as an entropy-regularized optimal transport problem. Derived from the JKO proximal scheme by lifting to the space of couplings and relaxing via the KL chain rule, each ETD iteration reduces to a Sinkhorn computation. The resulting transport plan provides global coordination, guiding each particle to nearby high-density proposals and naturally preserving multimodal structure. ETD can operate entirely score-free, requiring only pointwise evaluations of the unnormalized target density. Experiments on variance-collapse diagnostics, Bayesian logistic regression, neural networks, and molecular Boltzmann distributions show that ETD matches or outperforms SVGD, AGF-SVGD, and SGLD, with the largest gains in high-dimensional and multimodal settings.