MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；The Institute of Statistical Mathematics（统计数学研究所）；University of Cambridge（剑桥大学）；University College London（伦敦大学学院）
Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, Arthur Gretton, Bernhard Schölkopf
A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an empirical average is the standard estimate for the true kernel mean. We show that this estimator can be improved via a well-known phenomenon in statistics called Stein's phenomenon. After consideration, our theoretical analysis reveals the existence of a wide class of estimators that are better than the standard. Focusing on a subset of this class, we propose efficient shrinkage estimators for the kernel mean. Empirical evaluations on several benchmark applications clearly demonstrate that the proposed estimators outperform the standard kernel mean estimator.
This paper proposes and studies a numerical method for approximation of posterior expectations based on interpolation with a Stein reproducing kernel. Finite-sample-size bounds on the approximation error are established for posterior distributions supported on a compact Riemannian manifold, and we relate these to a kernel Stein discrepancy (KSD). Moreover, we prove in our setting that the KSD is equivalent to Sobolev discrepancy and, in doing so, we completely characterise the convergence-determining properties of KSD. Our contribution is rooted in a novel combination of Stein's method, the th…
Imperial College London（帝国理工学院）；The Alan Turing Institute（艾伦·图灵研究所）；University of Edinburgh（爱丁堡大学）
Bayesian inference problems require sampling or approximating high-dimensional probability distributions. The focus of this paper is on the recently introduced Stein variational gradient descent methodology, a class of algorithms that rely on iterated steepest descent steps with respect to a reproducing kernel Hilbert space norm. This construction leads to interacting particle systems, the mean-field limit of which is a gradient flow on the space of probability distributions equipped with a certain geometrical structure. We leverage this viewpoint to shed some light on the convergence properti…
University of Bath（巴斯大学）；The Alan Turing Institute（艾伦·图灵研究所）；Monash University（莫纳什大学）；Massachusetts Institute of Technology（麻省理工学院）；Heidelberg University（海德堡大学）
Stein importance sampling is a widely applicable technique based on kernelized Stein discrepancy, which corrects the output of approximate sampling algorithms by reweighting the empirical distribution of the samples. A general analysis of this technique is conducted for the previously unconsidered setting where samples are obtained via the simulation of a Markov chain, and applies to an arbitrary underlying Polish space. We prove that Stein importance sampling yields consistent estimators for quantities related to a target distribution of interest by using samples obtained from a geometrically…
University of Electronic Science and Technology of China（电子科技大学）；Harbin Institute of Technology（哈尔滨工业大学）；Peng Cheng Lab（鹏城实验室）
Stein variational gradient descent (SVGD) and its variants have shown promising successes in approximate inference for complex distributions. In practice, we notice that the kernel used in SVGD-based methods has a decisive effect on the empirical performance. Radial basis function (RBF) kernel with median heuristics is a common choice in previous approaches, but unfortunately this has proven to be sub-optimal. Inspired by the paradigm of Multiple Kernel Learning (MKL), our solution to this flaw is using a combination of multiple kernels to approximate the optimal kernel, rather than a single o…
Non-parametric goodness-of-fit testing procedures based on kernel Stein discrepancies (KSD) are promising approaches to validate general unnormalised distributions in various scenarios. Existing works focused on studying kernel choices to boost test performances. However, the choices of (non-unique) Stein operators also have considerable effect on the test performances. Inspired by the standardisation technique that was originally developed to better derive approximation properties for normal distributions, we present a unifying framework, called standardisation-function kernel Stein discrepan…
University of Oxford（牛津大学）；The Alan Turing Institute（阿兰·图灵研究所）
Score-based kernelised Stein discrepancy (KSD) tests have emerged as a powerful tool for the goodness of fit tests, especially in high dimensions; however, the test performance may depend on the choice of kernels in an underlying reproducing kernel Hilbert space (RKHS). Here we assess the effect of RKHS choice for KSD tests of random networks models, developed for exponential random graph models (ERGMs) in Xu and Reinert (2021)and for synthetic graph generators in Xu and Reinert (2022). We investigate the power performance and the computational runtime of the test in different scenarios, inclu…
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 precond…
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 wi…
CREST（经济与统计研究中心）；ENSAE（法国国立统计与经济管理学院）；Institut Polytechnique de Paris（巴黎理工学院）；CAS（中国科学院）；MINES ParisTech（巴黎高等矿业学院）；CMAP（应用数学中心）；Ecole Polytechnique（巴黎综合理工学院）；CNRS（法国国家科学研究中心）；DMA（数学系）；Ecole Normale Supérieure（巴黎高等师范学院）
Among dissimilarities between probability distributions, the Kernel Stein Discrepancy (KSD) has received much interest recently. We investigate the properties of its Wasserstein gradient flow to approximate a target probability distribution $π$ on $\mathbb{R}^d$, known up to a normalization constant. This leads to a straightforwardly implementable, deterministic score-based method to sample from $π$, named KSD Descent, which uses a set of particles to approximate $π$. Remarkably, owing to a tractable loss function, KSD Descent can leverage robust parameter-free optimization schemes such as L-B…
We study the estimation of the parametric components of single and multiple index volatility models. Using the first- and second-order Stein's identities, we develop methods that are applicable for the estimation of the variance index in the high-dimensional setting requiring finite moment condition, which allows for heavy-tailed data. Our approach complements the existing literature in the low-dimensional setting, while relaxing the conditions on estimation, and provides a novel approach in the high-dimensional setting. We prove that the statistical rate of convergence of our variance index e…
Stanford University（斯坦福大学）；Preferred Networks；The Institute of Statistical Mathematics（统计数理研究所）
We formalize an equivalence between two popular methods for Bayesian inference: Stein variational gradient descent (SVGD) and black-box variational inference (BBVI). In particular, we show that BBVI corresponds precisely to SVGD when the kernel is the neural tangent kernel. Furthermore, we interpret SVGD and BBVI as kernel gradient flows; we do this by leveraging the recent perspective that views SVGD as a gradient flow in the space of probability distributions and showing that BBVI naturally motivates a Riemannian structure on that space. We observe that kernel gradient flow also describes dy…
Stanford University（斯坦福大学）；Microsoft Research New England（微软新英格兰研究院）
Approximate Markov chain Monte Carlo (MCMC) offers the promise of more rapid sampling at the cost of more biased inference. Since standard MCMC diagnostics fail to detect these biases, researchers have developed computable Stein discrepancy measures that provably determine the convergence of a sample to its target distribution. This approach was recently combined with the theory of reproducing kernels to define a closed-form kernel Stein discrepancy (KSD) computable by summing kernel evaluations across pairs of sample points. We develop a theory of weak convergence for KSDs based on Stein's me…
Measure transport underpins several recent algorithms for posterior approximation in the Bayesian context, wherein a transport map is sought to minimise the Kullback--Leibler divergence (KLD) from the posterior to the approximation. The KLD is a strong mode of convergence, requiring absolute continuity of measures and placing restrictions on which transport maps can be permitted. Here we propose to minimise a kernel Stein discrepancy (KSD) instead, requiring only that the set of transport maps is dense in an $L^2$ sense and demonstrating how this condition can be validated. The consistency of …
Controlling Moments with Kernel Stein Discrepancies
Newcastle University（纽卡斯尔大学）；University College London（伦敦大学学院）；Gatsby Computational Neuroscience Unit, UCL（伦敦大学学院盖茨比计算神经科学单元）；Microsoft Research New England（微软研究院新英格兰分部）
Copula-Stein Discrepancy: A Generator-Based Stein Operator for Archimedean Dependence
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana, USA（路易斯安那州拉法叶大学数学系）；Department of Statistics, formerly of Presidency College, Kolkata, India（加德满都大学统计系）
A Kernel Stein Test for Comparing Latent Variable Models
University College London（伦敦大学学院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google Research（谷歌研究院）；Microsoft Research New England（微软新英格兰研究院）；The Institute of Statistical Mathematics（统计数学研究所）
Improved Finite-Particle Convergence Rates for Stein Variational Gradient Descent
Department of Statistics and Operations Research, University of North Carolina, Chapel Hill（统计与运筹学系，北卡罗来纳大学 Chapel Hill 分校）；Department of Statistics, University of California, Davis（统计系，加州大学 Davis 分校）；Department of Statistics, University of Chicago（统计系，芝加哥大学）
Large-scale kernel approximation is an important problem in machine learning research. Approaches using random Fourier features have become increasingly popular [Rahimi and Recht, 2007], where kernel approximation is treated as empirical mean estimation via Monte Carlo (MC) or Quasi-Monte Carlo (QMC) integration [Yang et al., 2014]. A limitation of the current approaches is that all the features receive an equal weight summing to 1. In this paper, we propose a novel shrinkage estimator from "Stein effect", which provides a data-driven weighting strategy for random features and enjoys theoretic…
We are interested in gradient-based Explicit Generative Modeling where samples can be derived from iterative gradient updates based on an estimate of the score function of the data distribution. Recent advances in Stochastic Gradient Langevin Dynamics (SGLD) demonstrates impressive results with energy-based models on high-dimensional and complex data distributions. Stein Variational Gradient Descent (SVGD) is a deterministic sampling algorithm that iteratively transports a set of particles to approximate a given distribution, based on functional gradient descent that decreases the KL divergenc…
We consider estimating the parametric components of semi-parametric multiple index models in a high-dimensional and non-Gaussian setting. Such models form a rich class of non-linear models with applications to signal processing, machine learning and statistics. Our estimators leverage the score function based first and second-order Stein's identities and do not require the covariates to satisfy Gaussian or elliptical symmetry assumptions common in the literature. Moreover, to handle score functions and responses that are heavy-tailed, our estimators are constructed via carefully thresholding t…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Kyoto University（京都大学）；RIKEN AIP（理化学研究所AIP）
We address the problem of non-parametric multiple model comparison: given $l$ candidate models, decide whether each candidate is as good as the best one(s) or worse than it. We propose two statistical tests, each controlling a different notion of decision errors. The first test, building on the post selection inference framework, provably controls the number of best models that are wrongly declared worse (false positive rate). The second test is based on multiple correction, and controls the proportion of the models declared worse but are in fact as good as the best (false discovery rate). We …
Department of Computer Science, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算机科学系）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
We propose a novel distributed inference algorithm for continuous graphical models, by extending Stein variational gradient descent (SVGD) to leverage the Markov dependency structure of the distribution of interest. Our approach combines SVGD with a set of structured local kernel functions defined on the Markov blanket of each node, which alleviates the curse of high dimensionality and simultaneously yields a distributed algorithm for decentralized inference tasks. We justify our method with theoretical analysis and show that the use of local kernels can be viewed as a new type of localized ap…
We propose and analyse a novel nonparametric goodness of fit testing procedure for exchangeable exponential random graph models (ERGMs) when a single network realisation is observed. The test determines how likely it is that the observation is generated from a target unnormalised ERGM density. Our test statistics are derived from a kernel Stein discrepancy, a divergence constructed via Steins method using functions in a reproducing kernel Hilbert space, combined with a discrete Stein operator for ERGMs. The test is a Monte Carlo test based on simulated networks from the target ERGM. We show th…
Stein-Rule Shrinkage for Stochastic Gradient Estimation in High Dimensions
Department of Statistics Faculty of Mathematical Sciences（统计学系数学科学学院）；Division of Computer Science Department of Applied Mathematics Faculty of Mathematical Sciences（计算机科学系应用数学系数学科学学院）
Imperial College London（帝国理工学院）；University College London（伦敦大学学院）；University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）
When maximum likelihood estimation is infeasible, one often turns to score matching, contrastive divergence, or minimum probability flow to obtain tractable parameter estimates. We provide a unifying perspective of these techniques as minimum Stein discrepancy estimators, and use this lens to design new diffusion kernel Stein discrepancy (DKSD) and diffusion score matching (DSM) estimators with complementary strengths. We establish the consistency, asymptotic normality, and robustness of DKSD and DSM estimators, then derive stochastic Riemannian gradient descent algorithms for their efficient …
Uniform-in-time Propagation-of-Chaos for Stein Variational Gradient Descent
University of California, Davis（加州大学戴维斯分校）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；ENSAE Institut Polytechnique de Paris（巴黎理工学院ENSAE）
ETH Zürich（苏黎世联邦理工学院）；Humboldt University of Berlin（柏林洪堡大学）
Ensembles of deep neural networks have achieved great success recently, but they do not offer a proper Bayesian justification. Moreover, while they allow for averaging of predictions over several hypotheses, they do not provide any guarantees for their diversity, leading to redundant solutions in function space. In contrast, particle-based inference methods, such as Stein variational gradient descent (SVGD), offer a Bayesian framework, but rely on the choice of a kernel to measure the similarity between ensemble members. In this work, we study different SVGD methods operating in the weight spa…
University of Technology Sydney（悉尼科技大学）；Microsoft Research New England（微软新英格兰研究院）；Opendoor Labs, Inc.（Opendoor实验室公司）；University of Warwick（华威大学）；Imperial College London（帝国理工学院）；Alan Turing Institute（艾伦·图灵研究所）；Newcastle University（纽卡斯尔大学）
An important task in computational statistics and machine learning is to approximate a posterior distribution $p(x)$ with an empirical measure supported on a set of representative points $\{x_i\}_{i=1}^n$. This paper focuses on methods where the selection of points is essentially deterministic, with an emphasis on achieving accurate approximation when $n$ is small. To this end, we present `Stein Points'. The idea is to exploit either a greedy or a conditional gradient method to iteratively minimise a kernel Stein discrepancy between the empirical measure and $p(x)$. Our empirical results demon…
