Stein Variational Gradient Descent (SVGD), introduced in [LW16], is a deterministic interactingparticle method for sampling from a target probability measure π e V, only requiring access to V. In the mean-field and continuous-time limit, the distribution of particles converges to a flow (ρt) in the space of probability measures that solves a variant of the Fokker-Planck equation with a velocity field smoothed by weighted convolution with a positive definite kernel [LLN19]. This flow can be interpreted as the gradient flow of the relative entropy H( |π) with respect to a "kernelized" Wasserstein metric [Liu17]. The goal of this paper is to investigate the convergence of (ρt) towards π. To this end, we focus on the model case of Riesz kernels of order s on the d-dimensional torus Td. This is a family of translation-invariant kernels whose Fourier coefficients decay as |ξ| 2s. The parameter s hence directly controls the "smoothing strength" of the interaction; in particular, continuous kernels correspond to s > d/2, C1 kernels to s > (d+1)/2, and C2 kernels to s > (d+2)/2. What is known: qualitative weak convergence The starting point of convergence analyses is the energy dissipation formula [Liu17] d dt H(ρt|π) = Is(ρt|π), (1.1) Authors are listed in alphabetical order.

Stein variational gradient descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate given distributions, based on a gradient-based update constructed to optimally decrease the KL divergence within a function space. This paper develops the first theoretical analysis on SVGD. We establish that the empirical measures of the SVGD samples weakly converge to the target distribution, and show that the asymptotic behavior of SVGD is characterized by a nonlinear Fokker-Planck equation known as Vlasov equation in physics. We develop a geometric perspective that views SVGD as a gradient flow of the KL divergence functional under a new metric structure on the space of distributions induced by Stein operator.

Stein variational gradient descent (SVGD) is a non-parametric inference algorithm that evolves a set of particles to fit a given distribution of interest. We analyze the non-asymptotic properties of SVGD, showing that there exists a set of functions, which we call the Stein matching set, whose expectations are exactly estimated by any set of particles that satisfies the fixed point equation of SVGD. This set is the image of Stein operator applied on the feature maps of the positive definite kernel used in SVGD. Our results provide a theoretical framework for analyzing the properties of SVGD with different kernels, shedding insight into optimal kernel choice. In particular, we show that SVGD with linear kernels yields exact estimation of means and variances on Gaussian distributions, while random Fourier features enable probabilistic bounds for distributional approximation. Our results offer a refreshing view of the classical inference problem as fitting Stein's identity or solving the Stein equation, which may motivate more efficient algorithms.

Stein V ariational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm.

Although the population (i.e, infinite-particle) limit dynamics of SVGD is well characterized, its behavior in the finite-particle regime is far less understood.

See Liu & Wang (2016, 2018); Liu (2017) formoredetails.

One of the core problems of modern statistics and machine learning is to approximate difficultto-compute probability distributions.