Sparse variational Gaussian process (SVGP) methods are a common choice for non-conjugate Gaussian process inference because of their computational benefits. In this paper, we improve their computational efficiency by using a dual parameterization where each data example is assigned dual parameters, similarly to site parameters used in expectation propagation. Our dual parameterization speeds-up inference using natural gradient descent, and provides a tighter evidence lower bound for hyperparameter learning. The approach has the same memory cost as the current SVGP methods, but it is faster and more accurate.
University of Warsaw（华沙大学）；ETH Zurich（苏黎世联邦理工学院）；University of Amsterdam（阿姆斯特丹大学）；Google Research（谷歌研究院）；Imperial College London（帝国理工学院）；University of California, Irvine（加利福尼亚大学尔湾分校）；Microsoft Research（微软研究院）
Variational Bayesian Inference is a popular methodology for approximating posterior distributions over Bayesian neural network weights. Recent work developing this class of methods has explored ever richer parameterizations of the approximate posterior in the hope of improving performance. In contrast, here we share a curious experimental finding that suggests instead restricting the variational distribution to a more compact parameterization. For a variety of deep Bayesian neural networks trained using Gaussian mean-field variational inference, we find that the posterior standard deviations c…
Columbia University（哥伦比亚大学）；New York Genome Center（纽约基因组中心）
Brittle optimization has been observed to adversely impact model likelihoods for regression and VAEs when simultaneously fitting neural network mappings from a (random) variable onto the mean and variance of a dependent Gaussian variable. Previous works have bolstered optimization and improved likelihoods, but fail other basic posterior predictive checks (PPCs). Under the PPC framework, we propose critiques to test predictive mean and variance calibration and the predictive distribution's ability to generate sensible data. We find that our attractively simple solution, to treat heteroscedastic…
The literature about history matching is vast and despite the impressive number of methods proposed and the significant progresses reported in the last decade, conditioning reservoir models to dynamic data is still a challenging task. Ensemble-based methods are among the most successful and efficient techniques currently available for history matching. These methods are usually able to achieve reasonable data matches, especially if an iterative formulation is employed. However, they sometimes fail to preserve the geological realism of the model, which is particularly evident in reservoir with …
Criteo AI Lab（Criteo人工智能实验室）；Durham University（杜伦大学）
The combination of the re-parameterization trick with the use of variational auto-encoders has caused a sensation in Bayesian deep learning, allowing the training of realistic generative models of images and has considerably increased our ability to use scalable latent variable models. The re-parameterization trick is necessary for models in which no analytical variational bound is available and allows noisy gradients to be computed for arbitrary models. However, for certain standard output layers of a neural network, analytical bounds are available and the variational auto-encoder may be used…
Gatsby Computational Neuroscience Unit（盖茨比计算神经科学单元）；University College London（伦敦大学学院）
Sparse variational approximations allow for principled and scalable inference in Gaussian Process (GP) models. In settings where several GPs are part of the generative model, theses GPs are a posteriori coupled. For many applications such as regression where predictive accuracy is the quantity of interest, this coupling is not crucial. Howewer if one is interested in posterior uncertainty, it cannot be ignored. A key element of variational inference schemes is the choice of the approximate posterior parameterization. When the number of latent variables is large, mean field (MF) methods provide…
Generalized additive models (GAMs) are a widely used class of models of interest to statisticians as they provide a flexible way to design interpretable models of data beyond linear models. We here propose a scalable and well-calibrated Bayesian treatment of GAMs using Gaussian processes (GPs) and leveraging recent advances in variational inference. We use sparse GPs to represent each component and exploit the additive structure of the model to efficiently represent a Gaussian a posteriori coupling between the components.
Tsinghua University（清华大学）；University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
Variational autoencoders (VAEs) combine latent variables with amortized variational inference, whose optimization usually converges into a trivial local optimum termed posterior collapse, especially in text modeling. By tracking the optimization dynamics, we observe the encoder-decoder incompatibility that leads to poor parameterizations of the data manifold. We argue that the trivial local optimum may be avoided by improving the encoder and decoder parameterizations since the posterior network is part of a transition map between them. To this end, we propose Coupled-VAE, which couples a VAE m…
University of Stuttgart（斯图加特大学）；ETAS GmbH（ETAS有限公司）
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems, so called recurrent models are frequently used. In this paper we introduce several new Deep recurrent Gaussian process (DRGP) models based on the Sparse Spectrum Gaussian process (SSGP) and the improved version, called variational Sparse Spectrum Gaussian process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific variational sparse Nyström approximation, the recurrent Gaussian pro…
Interactive Digital Media Institute（互动数字媒体研究所）；National University of Singapore（新加坡国立大学）
While much research effort has been dedicated to scaling up sparse Gaussian process (GP) models based on inducing variables for big data, little attention is afforded to the other less explored class of low-rank GP approximations that exploit the sparse spectral representation of a GP kernel. This paper presents such an effort to advance the state of the art of sparse spectrum GP models to achieve competitive predictive performance for massive datasets. Our generalized framework of stochastic variational Bayesian sparse spectrum GP (sVBSSGP) models addresses their shortcomings by adopting a Ba…
National University of Singapore（新加坡国立大学）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；MIT（麻省理工学院）
This paper presents a novel variational inference framework for deriving a family of Bayesian sparse Gaussian process regression (SGPR) models whose approximations are variationally optimal with respect to the full-rank GPR model enriched with various corresponding correlation structures of the observation noises. Our variational Bayesian SGPR (VBSGPR) models jointly treat both the distributions of the inducing variables and hyperparameters as variational parameters, which enables the decomposability of the variational lower bound that in turn can be exploited for stochastic optimization. Such…
The use of Gaussian process models is typically limited to datasets with a few tens of thousands of observations due to their complexity and memory footprint. The two most commonly used methods to overcome this limitation are 1) the variational sparse approximation which relies on inducing points and 2) the state-space equivalent formulation of Gaussian processes which can be seen as exploiting some sparsity in the precision matrix. We propose to take the best of both worlds: we show that the inducing point framework is still valid for state space models and that it can bring further computati…
Sparse, knot-based Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Certain sparse models can be derived through specific variational approximations to the true posterior, and knots can be selected to minimize the Kullback-Leibler divergence between the approximate and true posterior. While this has been a successful approach, simultaneous optimization of knots can be slow due to the number of parameters being optimized. Furthermore, there have been few proposed methods for selecting the number of knots, and no experimental results exi…
An Analysis of Posterior Collapse, Parameterization and Initialization in Variational Deep Gaussian Processes
Department of Computer Science and Artificial Intelligence（计算机科学与人工智能系）；Universidad de Granada（格拉纳达大学）；Department of Quantitative Methods（定量方法系）；CUNEF Universidad Madrid, Spain（CUNEF马德里大学, 西班牙）；Department of Computer Science and Centro de Investigación Avanzada en Física Fundamental（计算机科学系和先进物理基础研究中心）；Universidad Autónoma de Madrid（马德里自治大学）
Variational inference offers scalable and flexible tools to tackle intractable Bayesian inference of modern statistical models like Bayesian neural networks and Gaussian processes. For largely over-parameterized models, however, the over-regularization property of the variational objective makes the application of variational inference challenging. Inspired by the literature on kernel methods, and in particular on structured approximations of distributions of random matrices, this paper proposes Walsh-Hadamard Variational Inference, which uses Walsh-Hadamard-based factorization strategies to r…
Sparse coding strategies have been lauded for their parsimonious representations of data that leverage low dimensional structure. However, inference of these codes typically relies on an optimization procedure with poor computational scaling in high-dimensional problems. For example, sparse inference in the representations learned in the high-dimensional intermediary layers of deep neural networks (DNNs) requires an iterative minimization to be performed at each training step. As such, recent, quick methods in variational inference have been proposed to infer sparse codes by learning a distrib…
With a principled representation of uncertainty and closed form posterior updates, Gaussian processes (GPs) are a natural choice for online decision making. However, Gaussian processes typically require at least $\mathcal{O}(n^2)$ computations for $n$ training points, limiting their general applicability. Stochastic variational Gaussian processes (SVGPs) can provide scalable inference for a dataset of fixed size, but are difficult to efficiently condition on new data. We propose online variational conditioning (OVC), a procedure for efficiently conditioning SVGPs in an online setting that does…
National University of Singapore（新加坡国立大学）；Shenzhen University（深圳大学）
This paper presents a variational Bayesian kernel selection (VBKS) algorithm for sparse Gaussian process regression (SGPR) models. In contrast to existing GP kernel selection algorithms that aim to select only one kernel with the highest model evidence, our proposed VBKS algorithm considers the kernel as a random variable and learns its belief from data such that the uncertainty of the kernel can be interpreted and exploited to avoid overconfident GP predictions. To achieve this, we represent the probabilistic kernel as an additional variational variable in a variational inference (VI) framewo…
Monash University（莫纳什大学）；Indian Institute of Technology, Kanpur（印度理工学院坎普尔分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Many applications, such as text modelling, high-throughput sequencing, and recommender systems, require analysing sparse, high-dimensional, and overdispersed discrete (count-valued or binary) data. Although probabilistic matrix factorisation and linear/nonlinear latent factor models have enjoyed great success in modelling such data, many existing models may have inferior modelling performance due to the insufficient capability of modelling overdispersion in count-valued data and model misspecification in general. In this paper, we comprehensively study these issues and propose a variational au…
University of Cambridge（剑桥大学）；ETH Zürich（苏黎世联邦理工学院）；Warwick University（华威大学）
Large, multi-dimensional spatio-temporal datasets are omnipresent in modern science and engineering. An effective framework for handling such data are Gaussian process deep generative models (GP-DGMs), which employ GP priors over the latent variables of DGMs. Existing approaches for performing inference in GP-DGMs do not support sparse GP approximations based on inducing points, which are essential for the computational efficiency of GPs, nor do they handle missing data -- a natural occurrence in many spatio-temporal datasets -- in a principled manner. We address these shortcomings with the de…
Gaussian processes (GPs) are a powerful tool for probabilistic inference over functions. They have been applied to both regression and non-linear dimensionality reduction, and offer desirable properties such as uncertainty estimates, robustness to over-fitting, and principled ways for tuning hyper-parameters. However the scalability of these models to big datasets remains an active topic of research. We introduce a novel re-parametrisation of variational inference for sparse GP regression and latent variable models that allows for an efficient distributed algorithm. This is done by exploiting …
The Australian National University（澳大利亚国立大学）；Data61 CSIRO（联邦科学与工业研究组织数据61所）；University of Technology Sydney（悉尼科技大学）
The Hawkes process (HP) has been widely applied to modeling self-exciting events including neuron spikes, earthquakes and tweets. To avoid designing parametric triggering kernel and to be able to quantify the prediction confidence, the non-parametric Bayesian HP has been proposed. However, the inference of such models suffers from unscalability or slow convergence. In this paper, we aim to solve both problems. Specifically, first, we propose a new non-parametric Bayesian HP in which the triggering kernel is modeled as a squared sparse Gaussian process. Then, we propose a novel variational infe…
Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems so called recurrent models are used. In this article we introduce two new Deep Recurrent Gaussian Process (DRGP) models based on the Sparse Spectrum Gaussian Process (SSGP) and the improved variational version called Variational Sparse Spectrum Gaussian Process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific sparse Nyström approximation. Therefore, we also variationally integrat…
The natural gradient method has been used effectively in conjugate Gaussian process models, but the non-conjugate case has been largely unexplored. We examine how natural gradients can be used in non-conjugate stochastic settings, together with hyperparameter learning. We conclude that the natural gradient can significantly improve performance in terms of wall-clock time. For ill-conditioned posteriors the benefit of the natural gradient method is especially pronounced, and we demonstrate a practical setting where ordinary gradients are unusable. We show how natural gradients can be computed e…
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
In a variety of disciplines such as social sciences, psychology, medicine and economics, the recorded data are considered to be noisy measurements of latent variables connected by some causal structure. This corresponds to a family of graphical models known as the structural equation model with latent variables. While linear non-Gaussian variants have been well-studied, inference in nonparametric structural equation models is still underdeveloped. We introduce a sparse Gaussian process parameterization that defines a non-linear structure connecting latent variables, unlike common formulations …
Dalle Molle Institute for Artificial Intelligence (IDSIA)（Dalle Molle人工智能研究所 (IDSIA)）；Università della Svizzera Italiana (USI)（瑞士意大利语区大学 (USI)）；University of Limerick (UL)（利默里克大学 (UL)）
Sparse inducing points have long been a standard method to fit Gaussian processes to big data. In the last few years, spectral methods that exploit approximations of the covariance kernel have shown to be competitive. In this work we exploit a recently introduced orthogonally decoupled variational basis to combine spectral methods and sparse inducing points methods. We show that the method is competitive with the state-of-the-art on synthetic and on real-world data.
Non-negative tensor factorization models enable predictive analysis on count data. Among them, Bayesian Poisson-Gamma models can derive full posterior distributions of latent factors and are less sensitive to sparse count data. However, current inference methods for these Bayesian models adopt restricted update rules for the posterior parameters. They also fail to share the update information to better cope with the data sparsity. Moreover, these models are not endowed with a component that handles the imbalance in count data values. In this paper, we propose a novel variational auto-encoder f…
Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) for such zero-inflated datasets, produced by sparse kernels through learning a latent probit Gaussian process that can zero out kernel rows and columns whenever the signal is absent. The ZiGPs are particularly useful for making the powerful Gaussian process networks more interpre…
Excellent variational approximations to Gaussian process posteriors have been developed which avoid the $\mathcal{O}\left(N^3\right)$ scaling with dataset size $N$. They reduce the computational cost to $\mathcal{O}\left(NM^2\right)$, with $M\ll N$ being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in $N$, the true complexity of the algorithm depends on how $M$ must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We sho…
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
In a variety of disciplines such as social sciences, psychology, medicine and economics, the recorded data are considered to be noisy measurements of latent variables connected by some causal structure. This corresponds to a family of graphical models known as the structural equation model with latent variables. While linear non-Gaussian variants have been well-studied, inference in nonparametric structural equation models is still underdeveloped. We introduce a sparse Gaussian process parameterization that defines a non-linear structure connecting latent variables, unlike common formulations …
Nonnegative matrix factorization (NMF) with group sparsity constraints is formulated as a probabilistic graphical model and, assuming some observed data have been generated by the model, a feasible variational Bayesian algorithm is derived for learning model parameters. When used in a supervised learning scenario, NMF is most often utilized as an unsupervised feature extractor followed by classification in the obtained feature subspace. Having mapped the class labels to a more general concept of groups which underlie sparsity of the coefficients, what the proposed group sparse NMF model allows…
Introduction to Stochastic Differential Equations for Generative Machine Learning: A Variational Perspective
Department of Biology, University of Copenhagen（哥本哈根大学生物系）；Department of Applied Mathematics and Computer Science, Technical University of Denmark（丹麦技术大学应用数学与计算机科学系）；Google DeepMind（谷歌DeepMind）；Technical University of Berlin (TU Berlin)（柏林工业大学）；Technical University of Munich（慕尼黑工业大学）
Eurecom（欧洲通信学院）；National University of Singapore（新加坡国立大学）
Approximations to Gaussian processes based on inducing variables, combined with variational inference techniques, enable state-of-the-art sparse approaches to infer GPs at scale through mini batch-based learning. In this work, we address one limitation of sparse GPs, which is due to the challenge in dealing with a large number of inducing variables without imposing a special structure on the inducing inputs. In particular, we introduce a novel hierarchical prior, which imposes sparsity on the set of inducing variables. We treat our model variationally, and we experimentally show considerable c…
The variational framework for learning inducing variables (Titsias, 2009a) has had a large impact on the Gaussian process literature. The framework may be interpreted as minimizing a rigorously defined Kullback-Leibler divergence between the approximating and posterior processes. To our knowledge this connection has thus far gone unremarked in the literature. In this paper we give a substantial generalization of the literature on this topic. We give a new proof of the result for infinite index sets which allows inducing points that are not data points and likelihoods that depend on all functio…
Interest in multioutput kernel methods is increasing, whether under the guise of multitask learning, multisensor networks or structured output data. From the Gaussian process perspective a multioutput Mercer kernel is a covariance function over correlated output functions. One way of constructing such kernels is based on convolution processes (CP). A key problem for this approach is efficient inference. Alvarez and Lawrence (2009) recently presented a sparse approximation for CPs that enabled efficient inference. In this paper, we extend this work in two directions: we introduce the concept of…
State University of New York at Buffalo（纽约州立大学布法罗分校）
Learning with kernels is an important concept in machine learning. Standard approaches for kernel methods often use predefined kernels that require careful selection of hyperparameters. To mitigate this burden, we propose in this paper a framework to construct and learn a data-dependent kernel based on random features and implicit spectral distributions that are parameterized by deep neural networks. The constructed network (called KernelNet) can be applied to deep generative modeling in various scenarios, including two popular learning paradigms in deep generative models, MMD-GAN and implicit…
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the…
Ostwestfalen-Lippe University of Applied Sciences（东威斯特伐利亚-利珀应用科学大学）；Phoenix Contact Electronics GmbH（菲尼克斯电气电子有限公司）
Digitalization leads to data transparency for production systems that we can benefit from with data-driven analysis methods like neural networks. For example, automated anomaly detection enables saving resources and optimizing the production. We study using rarely occurring information about labeled anomalies into Variational Autoencoder neural network structures to overcome information deficits of supervised and unsupervised approaches. This method outperforms all other models in terms of accuracy, precision, and recall. We evaluate the following methods: Principal Component Analysis, Isolati…
Variational autoencoders (VAEs) often suffer from posterior collapse, which is a phenomenon in which the learned latent space becomes uninformative. This is often related to the hyperparameter resembling the data variance. It can be shown that an inappropriate choice of this hyperparameter causes the oversmoothness in the linearly approximated case and can be empirically verified for the general cases. Moreover, determining such appropriate choice becomes infeasible if the data variance is non-uniform or conditional. Therefore, we propose VAE extensions with generalized parameterizations of th…
We consider the problem of learning a Gaussian variational approximation to the posterior distribution for a high-dimensional parameter, where we impose sparsity in the precision matrix to reflect appropriate conditional independence structure in the model. Incorporating sparsity in the precision matrix allows the Gaussian variational distribution to be both flexible and parsimonious, and the sparsity is achieved through parameterization in terms of the Cholesky factor. Efficient stochastic gradient methods which make appropriate use of gradient information for the target distribution are deve…
We present a new flavor of Variational Autoencoder (VAE) that interpolates seamlessly between unsupervised, semi-supervised and fully supervised learning domains. We show that unlabeled datapoints not only boost unsupervised tasks, but also the classification performance. Vice versa, every label not only improves classification, but also unsupervised tasks. The proposed architecture is simple: A classification layer is connected to the topmost encoder layer, and then combined with the resampled latent layer for the decoder. The usual evidence lower bound (ELBO) loss is supplemented with a supe…
Paul G. Allen School of Computer Science & Engineering（保罗·G·艾伦计算机科学与工程学院）；University of Washington（华盛顿大学）
Learning a generative model of visual information with sparse and compositional features has been a challenge for both theoretical neuroscience and machine learning communities. Sparse coding models have achieved great success in explaining the receptive fields of mammalian primary visual cortex with sparsely activated latent representation. In this paper, we focus on a recently proposed model, sparse coding variational autoencoder (SVAE) (Barello et al., 2018), and show that the end-to-end training scheme of SVAE leads to a large group of decoding filters not fully optimized with noise-like r…
