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.
Department of Biostatistics, Harvard（哈佛大学生物统计学系）；Microsoft Research New England（微软新英格兰研究院）
Computable Stein discrepancies have been deployed for a variety of applications, ranging from sampler selection in posterior inference to approximate Bayesian inference to goodness-of-fit testing. Existing convergence-determining Stein discrepancies admit strong theoretical guarantees but suffer from a computational cost that grows quadratically in the sample size. While linear-time Stein discrepancies have been proposed for goodness-of-fit testing, they exhibit avoidable degradations in testing power -- even when power is explicitly optimized. To address these shortcomings, we introduce featu…
Much of machine learning relies on comparing distributions with discrepancy measures. Stein's method creates discrepancy measures between two distributions that require only the unnormalized density of one and samples from the other. Stein discrepancies can be combined with kernels to define kernelized Stein discrepancies (KSDs). While kernels make Stein discrepancies tractable, they pose several challenges in high dimensions. We introduce kernelized complete conditional Stein discrepancies (KCC-SDs). Complete conditionals turn a multivariate distribution into multiple univariate distributions…
Stein discrepancies have emerged as a powerful statistical tool, being applied to fundamental statistical problems including parameter inference, goodness-of-fit testing, and sampling. The canonical Stein discrepancies require the derivatives of a statistical model to be computed, and in return provide theoretical guarantees of convergence detection and control. However, for complex statistical models, the stable numerical computation of derivatives can require bespoke algorithmic development and render Stein discrepancies impractical. This paper focuses on posterior approximation using Stein …
Performance guarantees for compression in nonlinear models under non-Gaussian observations can be achieved through the use of distributional characteristics that are sensitive to the distance to normality, and which in particular return the value of zero under Gaussian or linear sensing. The use of these characteristics, or discrepancies, improves some previous results in this area by relaxing conditions and tightening performance bounds. In addition, these characteristics are tractable to compute when Gaussian sensing is corrupted by either additive errors or mixing.
Queensland University of Technology（昆士兰科技大学）；Centre for Data Science, Queensland University of Technology（昆士兰科技大学数据科学中心）；Quantium；ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems (MACSYS)（ARC细胞系统数学分析卓越中心）
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）
We propose in this work RBM-SVGD, a stochastic version of Stein Variational Gradient Descent (SVGD) method for efficiently sampling from a given probability measure and thus useful for Bayesian inference. The method is to apply the Random Batch Method (RBM) for interacting particle systems proposed by Jin et al to the interacting particle systems in SVGD. While keeping the behaviors of SVGD, it reduces the computational cost, especially when the interacting kernel has long range. Numerical examples verify the efficiency of this new version of SVGD.
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 Cambridge（剑桥大学）；Microsoft Research Cambridge（微软研究院剑桥分部）；Imperial College London（帝国理工学院）；The Alan Turing Institute（阿兰·图灵研究所）
Kernelized Stein discrepancy (KSD), though being extensively used in goodness-of-fit tests and model learning, suffers from the curse-of-dimensionality. We address this issue by proposing the sliced Stein discrepancy and its scalable and kernelized variants, which employ kernel-based test functions defined on the optimal one-dimensional projections. When applied to goodness-of-fit tests, extensive experiments show the proposed discrepancy significantly outperforms KSD and various baselines in high dimensions. For model learning, we show its advantages over existing Stein discrepancy baselines …
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Stanford University（斯坦福大学）
Stein variational gradient descent (SVGD) is a general-purpose optimization-based sampling algorithm that has recently exploded in popularity, but is limited by two issues: it is known to produce biased samples, and it can be slow to converge on complicated distributions. A recently proposed stochastic variant of SVGD (sSVGD) addresses the first issue, producing unbiased samples by incorporating a special noise into the SVGD dynamics such that asymptotic convergence is guaranteed. Meanwhile, Stein variational Newton (SVN), a Newton-like extension of SVGD, dramatically accelerates the convergen…
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）
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…
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…
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 …
Aalto University（阿尔托大学）；Department of Computer Science（计算机科学系）
Approximate Bayesian inference estimates descriptors of an intractable target distribution - in essence, an optimization problem within a family of distributions. For example, Langevin dynamics (LD) extracts asymptotically exact samples from a diffusion process because the time evolution of its marginal distributions constitutes a curve that minimizes the KL-divergence via steepest descent in the Wasserstein space. Parallel to LD, Stein variational gradient descent (SVGD) similarly minimizes the KL, albeit endowed with a novel Stein-Wasserstein distance, by deterministically transporting a set…
Institute of Mathematics, Universität Potsdam（波茨坦大学数学研究所）；WIAS Berlin（柏林应用分析与随机学研究所）
Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: variational inference and Markov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent …
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…
The Polynomial Stein Discrepancy for Assessing Moment Convergence
School of Mathematical Sciences, Queensland University of Technology, Australia（昆士兰技术大学数学科学学院，澳大利亚）；Centre for Data Science, Queensland University of Technology, Australia（昆士兰技术大学数据科学中心，澳大利亚）；School of Mathematics and Physics, University of Queensland, Brisbane, Australia（昆士兰大学数学与物理学院，布里斯班，澳大利亚）
Stanford University（斯坦福大学）；Imperial College London（帝国理工学院）；University of Warwick（华威大学）；Microsoft Research New England（微软新英格兰研究院）
Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Ito diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Ito diffusion. As example applications, we develop computable and convergence-determining…
Department of Statistics, University of California, Davis（加利福尼亚大学戴维斯分校统计系）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high-dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an…
Institute of Mathematical Sciences (ICMAT)（数学科学研究所（ICMAT））；SAMSI, Duke University（杜克大学统计与应用数学研究所（SAMSI））；Duke University（杜克大学）
We propose a unifying view of two different Bayesian inference algorithms, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) and Stein Variational Gradient Descent (SVGD), leading to improved and efficient novel sampling schemes. We show that SVGD combined with a noise term can be framed as a multiple chain SG-MCMC method. Instead of treating each parallel chain independently from others, our proposed algorithm implements a repulsive force between particles, avoiding collapse and facilitating a better exploration of the parameter space. We also show how the addition of this noise term is …
Fudan University（复旦大学）；SUNY at Buffalo（纽约州立大学布法罗分校）
Stochastic particle-optimization sampling (SPOS) is a recently-developed scalable Bayesian sampling framework that unifies stochastic gradient MCMC (SG-MCMC) and Stein variational gradient descent (SVGD) algorithms based on Wasserstein gradient flows. With a rigorous non-asymptotic convergence theory developed recently, SPOS avoids the particle-collapsing pitfall of SVGD. Nevertheless, variance reduction in SPOS has never been studied. In this paper, we bridge the gap by presenting several variance-reduction techniques for SPOS. Specifically, we propose three variants of variance-reduced SPOS,…
State University of New York at Buffalo（纽约州立大学布法罗分校）
Recently, many machine learning and statistical models such as non-linear regressions, the Single Index, Multi-index, Varying Coefficient Index Models and Two-layer Neural Networks can be reduced to or be seen as a special case of a new model which is called the \textit{Stochastic Linear Combination of Non-linear Regressions} model. However, due to the high non-convexity of the problem, there is no previous work study how to estimate the model. In this paper, we provide the first study on how to estimate the model efficiently and scalably. Specifically, we first show that with some mild assump…
Stein Kernelized Molecular Dynamics for Active Learning of Interatomic Potentials
Center for Computational Science & Engineering, Schwarzman College of Computing, MIT（计算科学与工程中心，计算机科学学院，麻省理工学院）；Warwick Centre for Predictive Modelling, School of Engineering, University of Warwick（预测建模中心，工程学院，沃里克大学）；NVIDIA
We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of convergence of a multivariate martingale difference sequence to a normal random vector, which might be of independent interest. We obtain the explicit rates for the multivariate martingale CLT using a combination of Stein's method and Lindeberg's argument, which is then used in conjun…
Duke University（杜克大学）；University at Buffalo, SUNY（纽约州立大学布法罗分校）
Particle-optimization-based sampling (POS) is a recently developed effective sampling technique that interactively updates a set of particles. A representative algorithm is the Stein variational gradient descent (SVGD). We prove, under certain conditions, SVGD experiences a theoretical pitfall, {\it i.e.}, particles tend to collapse. As a remedy, we generalize POS to a stochastic setting by injecting random noise into particle updates, thus yielding particle-optimization sampling (SPOS). Notably, for the first time, we develop {\em non-asymptotic convergence theory} for the SPOS framework (rel…
Statistical inference for a stochastic wave equation with Malliavin calculus
Conacyt Research Fellow - Universidad Nacional Autónoma de México. Instituto de Matemáticas（康赛特研究员 - 墨西哥国立自治大学。数学研究所）；Facultad de ciencias - Universidad Nacional Autónoma de México（理学院 - 墨西哥国立自治大学）
We propose a simple algorithm to train stochastic neural networks to draw samples from given target distributions for probabilistic inference. Our method is based on iteratively adjusting the neural network parameters so that the output changes along a Stein variational gradient direction (Liu & Wang, 2016) that maximally decreases the KL divergence with the target distribution. Our method works for any target distribution specified by their unnormalized density function, and can train any black-box architectures that are differentiable in terms of the parameters we want to adapt. We demonstra…
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…
Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation of the gradient of the potential function. In several real-world scenarios, obtaining gradient evaluations might either be computationally expensive, or simply impossible. In this work, we propose and analyze stochastic zeroth-order sampling algorithms for discretizing overdamped and underdamped Langevin diffusions. Our approach is based on estimating the gradients, based on Gaussian Stein's identities, widely used in the stochastic optimization…
