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.

There have been extensive research trying to utilize the hidden geometric information of data samples [1, 2, 3].

Stein Variational Gradient Descent (SVGD) is a popular particle-based variational inference algorithm with impressive empirical performance across various domains. 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. To this end, our work introduces the notion of *virtual particles* to develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, that are exactly realizable using finite particles.

Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SVGD updates. We achieve this by presenting a generalization of SVGD that replaces the scalar-valued kernels in vanilla SVGD with more general matrix-valued kernels. This yields a significant extension of SVGD, and more importantly, allows us to flexibly incorporate various preconditioning matricesto accelerate the exploration in the probability landscape. Empirical results show that our method outperforms vanilla SVGD and a variety of baseline approaches over a range of real-world Bayesian inference tasks.

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.

We provide the first finite-particle convergence rate for Stein variational gradient descent (SVGD), a popular algorithm for approximating a probability distribution with a collection of particles. Specifically, whenever the target distribution is sub-Gaussian with a Lipschitz score, SVGD with $n$ particles and an appropriate step size sequence drives the kernel Stein discrepancy to zero at an order ${1/}{\sqrt{\log\log n}}$ rate. We suspect that the dependence on $n$ can be improved, and we hope that our explicit, non-asymptotic proof strategy will serve as a template for future refinements.

Sampling-based inference and learning techniques, especially Bayesian inference, provide an essential approach to handling uncertainty in machine learning (ML). As these techniques are increasingly used in daily life, it becomes essential to safeguard the ML systems with various trustworthy-related constraints, such as fairness, safety, interpretability. Mathematically, enforcing these constraints in probabilistic inference can be cast into sampling from intractable distributions subject to general nonlinear constraints, for which practical efficient algorithms are still largely missing. In this work, we propose a family of constrained sampling algorithms which generalize Langevin Dynamics (LD) and Stein Variational Gradient Descent (SVGD) to incorporate a moment constraint specified by a general nonlinear function. By exploiting the gradient flow structure of LD and SVGD, we derive two types of algorithms for handling constraints, including a primal-dual gradient approach and the constraint controlled gradient descent approach. We investigate the continuous-time mean-field limit of these algorithms and show that they have O(1/t) convergence under mild conditions. Moreover, the LD variant converges linearly assuming that a log Sobolev like inequality holds. Various numerical experiments are conducted to demonstrate the efficiency of our algorithms in trustworthy settings.

Stein discrepancies (SDs) monitor convergence and non-convergence in approximate inference when exact integration and sampling are intractable. However, the computation of a Stein discrepancy can be prohibitive if the Stein operator -- often a sum over likelihood terms or potentials -- is expensive to evaluate. To address this deficiency, we show that stochastic Stein discrepancies (SSDs) based on subsampled approximations of the Stein operator inherit the convergence control properties of standard SDs with probability 1. Along the way, we establish the convergence of Stein variational gradient descent (SVGD) on unbounded domains, resolving an open question of Liu (2017). In our experiments with biased Markov chain Monte Carlo (MCMC) hyperparameter tuning, approximate MCMC sampler selection, and stochastic SVGD, SSDs deliver comparable inferences to standard SDs with orders of magnitude fewer likelihood evaluations.