Gaussian process (GP) regression with 1D inputs can often be performed in linear time via a stochastic differential equation formulation. However, for non-Gaussian likelihoods, this requires application of approximate inference methods which can make the implementation difficult, e.g., expectation propagation can be numerically unstable and variational inference can be computationally inefficient. In this paper, we propose a new method that removes such difficulties. Building upon an existing method called conjugate-computation variational inference, our approach enables linear-time inference via Kalman recursions while avoiding numerical instabilities and convergence issues. We provide an efficient JAX implementation which exploits just-in-time compilation and allows for fast automatic differentiation through large for-loops. Overall, our approach leads to fast and stable variational inference in state-space GP models that can be scaled to time series with millions of data points.
Deep Gaussian processes (DGPs) can model complex marginal densities as well as complex mappings. Non-Gaussian marginals are essential for modelling real-world data, and can be generated from the DGP by incorporating uncorrelated variables to the model. Previous work on DGP models has introduced noise additively and used variational inference with a combination of sparse Gaussian processes and mean-field Gaussians for the approximate posterior. Additive noise attenuates the signal, and the Gaussian form of variational distribution may lead to an inaccurate posterior. We instead incorporate nois…
Georgia Institute of Technology（佐治亚理工学院）；Institute for Robotics Intelligent Machines（机器人与智能机器研究所）
Large-scale Gaussian process inference has long faced practical challenges due to time and space complexity that is superlinear in dataset size. While sparse variational Gaussian process models are capable of learning from large-scale data, standard strategies for sparsifying the model can prevent the approximation of complex functions. In this work, we propose a novel variational Gaussian process model that decouples the representation of mean and covariance functions in reproducing kernel Hilbert space. We show that this new parametrization generalizes previous models. Furthermore, it yields…
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.
Pohang University of Science and Technology（浦项科技大学）
The variational autoencoder (VAE) is a generative model with continuous latent variables where a pair of probabilistic encoder (bottom-up) and decoder (top-down) is jointly learned by stochastic gradient variational Bayes. We first elaborate Gaussian VAE, approximating the local covariance matrix of the decoder as an outer product of the principal direction at a position determined by a sample drawn from Gaussian distribution. We show that this model, referred to as VAE-ROC, better captures the data manifold, compared to the standard Gaussian VAE where independent multivariate Gaussian was use…
We develop a scalable deep non-parametric generative model by augmenting deep Gaussian processes with a recognition model. Inference is performed in a novel scalable variational framework where the variational posterior distributions are reparametrized through a multilayer perceptron. The key aspect of this reformulation is that it prevents the proliferation of variational parameters which otherwise grow linearly in proportion to the sample size. We derive a new formulation of the variational lower bound that allows us to distribute most of the computation in a way that enables to handle datas…
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…
University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦·图灵研究所）
Variational autoencoders (VAEs) are a powerful class of deep generative latent variable model for unsupervised representation learning on high-dimensional data. To ensure computational tractability, VAEs are often implemented with a univariate standard Gaussian prior and a mean-field Gaussian variational posterior distribution. This results in a vector-valued latent variables that are agnostic to the original data structure which might be highly correlated across and within multiple dimensions. We propose a tensor-variate extension to the VAE framework, the tensor-variate Gaussian process prio…
University of Sheffield（谢菲尔德大学）；Department of Computer Science（计算机科学系）；Sheffield, UK（英国谢菲尔德）
This paper introduces a method to approximate Gaussian process regression by representing the problem as a stochastic differential equation and using variational inference to approximate solutions. The approximations are compared with full GP regression and generated paths are demonstrated to be indistinguishable from GP samples. We show that the approach extends easily to non-linear dynamics and discuss extensions to which the approach can be easily applied.
We show how to use Stein variational gradient descent (SVGD) to carry out inference in Gaussian process (GP) models with non-Gaussian likelihoods and large data volumes. Markov chain Monte Carlo (MCMC) is extremely computationally intensive for these situations, but the parametric assumptions required for efficient variational inference (VI) result in incorrect inference when they encounter the multi-modal posterior distributions that are common for such models. SVGD provides a non-parametric alternative to variational inference which is substantially faster than MCMC. We prove that for GP mod…
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…
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…
Aalto University（阿尔托大学）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
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…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Microsoft Research New England（微软新英格兰研究院）；New York University（纽约大学）
We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We propose the harmonic kernel decomposition (HKD), which uses Fourier series to decompose a kernel as a sum of orthogonal kernels. Our variational approximation exploits this orthogonality to enable a large number of inducing points at a low computational cost. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections, and it significantly …
New York University（纽约大学）；Facebook Inc.（脸书公司）；University of Sydney（悉尼大学）
Through sequential construction of posteriors on observing data online, Bayes' theorem provides a natural framework for continual learning. We develop Variational Auto-Regressive Gaussian Processes (VAR-GPs), a principled posterior updating mechanism to solve sequential tasks in continual learning. By relying on sparse inducing point approximations for scalable posteriors, we propose a novel auto-regressive variational distribution which reveals two fruitful connections to existing results in Bayesian inference, expectation propagation and orthogonal inducing points. Mean predictive entropy es…
ETH Zürich（苏黎世联邦理工学院）；University of Cambridge（剑桥大学）；University of Warwick（华威大学）；University of California, Irvine（加州大学欧文分校）
Conventional variational autoencoders fail in modeling correlations between data points due to their use of factorized priors. Amortized Gaussian process inference through GP-VAEs has led to significant improvements in this regard, but is still inhibited by the intrinsic complexity of exact GP inference. We improve the scalability of these methods through principled sparse inference approaches. We propose a new scalable GP-VAE model that outperforms existing approaches in terms of runtime and memory footprint, is easy to implement, and allows for joint end-to-end optimization of all components…
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…
Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models
LMU Munich（慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Imperial College London（帝国理工学院）；University of Technology Nuremberg (UTN)（纽伦堡工业大学）；Helmholtz AI（亥姆霍兹人工智能中心）；Xyme
Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
Deep Gaussian Processes (DGPs) are multi-layer, flexible extensions of Gaussian processes but their training remains challenging. Sparse approximations simplify the training but often require optimization over a large number of inducing inputs and their locations across layers. In this paper, we simplify the training by setting the locations to a fixed subset of data and sampling the inducing inputs from a variational distribution. This reduces the trainable parameters and computation cost without significant performance degradations, as demonstrated by our empirical results on regression prob…
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…
We utilize copulas to constitute a unified framework for constructing and optimizing variational proposals in hierarchical Bayesian models. For models with continuous and non-Gaussian hidden variables, we propose a semiparametric and automated variational Gaussian copula approach, in which the parametric Gaussian copula family is able to preserve multivariate posterior dependence, and the nonparametric transformations based on Bernstein polynomials provide ample flexibility in characterizing the univariate marginal posteriors.
Aalto University（阿尔托大学）；University of Tübingen（蒂宾根大学）
Recent machine learning advances have proposed black-box estimation of unknown continuous-time system dynamics directly from data. However, earlier works are based on approximative ODE solutions or point estimates. We propose a novel Bayesian nonparametric model that uses Gaussian processes to infer posteriors of unknown ODE systems directly from data. We derive sparse variational inference with decoupled functional sampling to represent vector field posteriors. We also introduce a probabilistic shooting augmentation to enable efficient inference from arbitrarily long trajectories. The method …
Bayesian methods have become a popular way to incorporate prior knowledge and a notion of uncertainty into machine learning models. At the same time, the complexity of modern machine learning makes it challenging to comprehend a model's reasoning process, let alone express specific prior assumptions in a rigorous manner. While primarily interested in the former issue, recent developments intransparent machine learning could also broaden the range of prior information that we can provide to complex Bayesian models. Inspired by the idea of self-explaining models, we introduce a corresponding con…
University of Utah（犹他大学）；School of Computing（计算学院）；Scientific Computing and Imaging Institute（科学计算与成像研究所）
Gaussian process regression networks (GPRN) are powerful Bayesian models for multi-output regression, but their inference is intractable. To address this issue, existing methods use a fully factorized structure (or a mixture of such structures) over all the outputs and latent functions for posterior approximation, which, however, can miss the strong posterior dependencies among the latent variables and hurt the inference quality. In addition, the updates of the variational parameters are inefficient and can be prohibitively expensive for a large number of outputs. To overcome these limitations…
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…
Aalto University（阿尔托大学）；Delft University of Technology（代尔夫特理工大学）
Gaussian processes (GPs) provide a powerful framework for extrapolation, interpolation, and noise removal in regression and classification. This paper considers constraining GPs to arbitrarily-shaped domains with boundary conditions. We solve a Fourier-like generalised harmonic feature representation of the GP prior in the domain of interest, which both constrains the GP and attains a low-rank representation that is used for speeding up inference. The method scales as $\mathcal{O}(nm^2)$ in prediction and $\mathcal{O}(m^3)$ in hyperparameter learning for regression, where $n$ is the number of …
Complex multivariate time series arise in many fields, ranging from computer vision to robotics or medicine. Often we are interested in the independent underlying factors that give rise to the high-dimensional data we are observing. While many models have been introduced to learn such disentangled representations, only few attempt to explicitly exploit the structure of sequential data. We investigate the disentanglement properties of Gaussian process variational autoencoders, a class of models recently introduced that have been successful in different tasks on time series data. Our model explo…
The Alan Turing Institute（艾伦·图灵研究所）；Aalto University（阿尔托大学）；NVIDIA（英伟达）；University of Warwick（华威大学）
We introduce a scalable approach to Gaussian process inference that combines spatio-temporal filtering with natural gradient variational inference, resulting in a non-conjugate GP method for multivariate data that scales linearly with respect to time. Our natural gradient approach enables application of parallel filtering and smoothing, further reducing the temporal span complexity to be logarithmic in the number of time steps. We derive a sparse approximation that constructs a state-space model over a reduced set of spatial inducing points, and show that for separable Markov kernels the full …
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…
Gaussian process classification is a popular method with a number of appealing properties. We show how to scale the model within a variational inducing point framework, outperforming the state of the art on benchmark datasets. Importantly, the variational formulation can be exploited to allow classification in problems with millions of data points, as we demonstrate in experiments.
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；UC Berkeley（加州大学伯克利分校）
This paper presents an efficient variational inference framework for deriving a family of structured gaussian process regression network (SGPRN) models. The key idea is to incorporate auxiliary inducing variables in latent functions and jointly treats both the distributions of the inducing variables and hyper-parameters as variational parameters. Then we propose structured variable distributions and marginalize latent variables, which enables the decomposability of a tractable variational lower bound and leads to stochastic optimization. Our inference approach is able to model data in which ou…
Variational autoencoders often assume isotropic Gaussian priors and mean-field posteriors, hence do not exploit structure in scenarios where we may expect similarity or consistency across latent variables. Gaussian process variational autoencoders alleviate this problem through the use of a latent Gaussian process, but lead to a cubic inference time complexity. We propose a more scalable extension of these models by leveraging the independence of the auxiliary features, which is present in many datasets. Our model factorizes the latent kernel across these features in different dimensions, lead…
We develop a framework for generalized variational inference in infinite-dimensional function spaces and use it to construct a method termed Gaussian Wasserstein inference (GWI). GWI leverages the Wasserstein distance between Gaussian measures on the Hilbert space of square-integrable functions in order to determine a variational posterior using a tractable optimisation criterion and avoids pathologies arising in standard variational function space inference. An exciting application of GWI is the ability to use deep neural networks in the variational parametrisation of GWI, combining their sup…
University of Amsterdam（阿姆斯特丹大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
We introduce a variational Bayesian neural network where the parameters are governed via a probability distribution on random matrices. Specifically, we employ a matrix variate Gaussian \cite{gupta1999matrix} parameter posterior distribution where we explicitly model the covariance among the input and output dimensions of each layer. Furthermore, with approximate covariance matrices we can achieve a more efficient way to represent those correlations that is also cheaper than fully factorized parameter posteriors. We further show that with the "local reprarametrization trick" \cite{kingma2015va…
Hidden Markov Models (HMM) have been used for several years in many time series analysis or pattern recognitions tasks. HMM are often trained by means of the Baum-Welch algorithm which can be seen as a special variant of an expectation maximization (EM) algorithm. Second-order training techniques such as Variational Bayesian Inference (VI) for probabilistic models regard the parameters of the probabilistic models as random variables and define distributions over these distribution parameters, hence the name of this technique. VI can also bee regarded as a special case of an EM algorithm. In th…
University of Oxford（牛津大学）；Imperial College London（帝国理工学院）
Variational Gaussian process (GP) approximations have become a standard tool in fast GP inference. This technique requires a user to select variational features to increase efficiency. So far the common choices in the literature are disparate and lacking generality. We propose to view the GP as lying in a Banach space which then facilitates a unified perspective. This is used to understand the relationship between existing features and to draw a connection between kernel ridge regression and variational GP approximations.
Deep Gaussian processes provide a flexible approach to probabilistic modelling of data using either supervised or unsupervised learning. For tractable inference approximations to the marginal likelihood of the model must be made. The original approach to approximate inference in these models used variational compression to allow for approximate variational marginalization of the hidden variables leading to a lower bound on the marginal likelihood of the model [Damianou and Lawrence, 2013]. In this paper we extend this idea with a nested variational compression. The resulting lower bound on the…
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…
University of California Santa Barbara（加州大学圣巴巴拉分校）
Variational language models seek to estimate the posterior of latent variables with an approximated variational posterior. The model often assumes the variational posterior to be factorized even when the true posterior is not. The learned variational posterior under this assumption does not capture the dependency relationships over latent variables. We argue that this would cause a typical training problem called posterior collapse observed in all other variational language models. We propose Gaussian Copula Variational Autoencoder (VAE) to avert this problem. Copula is widely used to model co…
Thanks to the reparameterization trick, deep latent Gaussian models have shown tremendous success recently in learning latent representations. The ability to couple them however with nonparamet-ric priors such as the Dirichlet Process (DP) hasn't seen similar success due to its non parameteriz-able nature. In this paper, we present an alternative treatment of the variational posterior of the Dirichlet Process Deep Latent Gaussian Mixture Model (DP-DLGMM), where we show that the prior cluster parameters and the variational posteriors of the beta distributions and cluster hidden variables can be…
We present the first framework for Gaussian-process-modulated Poisson processes when the temporal data appear in the form of panel counts. Panel count data frequently arise when experimental subjects are observed only at discrete time points and only the numbers of occurrences of the events between subsequent observation times are available. The exact occurrence timestamps of the events are unknown. The method of conducting the efficient variational inference is presented, based on the assumption of a Gaussian-process-modulated intensity function. We derive a tractable lower bound to alleviate…
Gaussian process latent variable models (GPLVM) are a flexible and non-linear approach to dimensionality reduction, extending classical Gaussian processes to an unsupervised learning context. The Bayesian incarnation of the GPLVM Titsias and Lawrence, 2010] uses a variational framework, where the posterior over latent variables is approximated by a well-behaved variational family, a factorized Gaussian yielding a tractable lower bound. However, the non-factories ability of the lower bound prevents truly scalable inference. In this work, we study the doubly stochastic formulation of the Bayesia…
Variational inference is a powerful tool for approximate inference, and it has been recently applied for representation learning with deep generative models. We develop the variational Gaussian process (VGP), a Bayesian nonparametric variational family, which adapts its shape to match complex posterior distributions. The VGP generates approximate posterior samples by generating latent inputs and warping them through random non-linear mappings; the distribution over random mappings is learned during inference, enabling the transformed outputs to adapt to varying complexity. We prove a universal…
Gaussian processes (GPs) are powerful non-parametric function estimators. However, their applications are largely limited by the expensive computational cost of the inference procedures. Existing stochastic or distributed synchronous variational inferences, although have alleviated this issue by scaling up GPs to millions of samples, are still far from satisfactory for real-world large applications, where the data sizes are often orders of magnitudes larger, say, billions. To solve this problem, we propose ADVGP, the first Asynchronous Distributed Variational Gaussian Process inference for reg…
Inference for GP models with non-Gaussian noises is computationally expensive when dealing with large datasets. Many recent inference methods approximate the posterior distribution with a simpler distribution defined on a small number of inducing points. The inference is accurate only when data points have strong correlation with these inducing points. In this paper, we consider the inference problem in a different direction: GP function values in the posterior are mostly correlated in short distance. We construct a variational distribution such that the inference for a data point considers on…
This work brings together two powerful concepts in Gaussian processes: the variational approach to sparse approximation and the spectral representation of Gaussian processes. This gives rise to an approximation that inherits the benefits of the variational approach but with the representational power and computational scalability of spectral representations. The work hinges on a key result that there exist spectral features related to a finite domain of the Gaussian process which exhibit almost-independent covariances. We derive these expressions for Matern kernels in one dimension, and genera…
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…
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）；USC Information Sciences Institute（南加州大学信息科学研究所）
Achieving the full promise of the Thermodynamic Variational Objective (TVO), a recently proposed variational lower bound on the log evidence involving a one-dimensional Riemann integral approximation, requires choosing a "schedule" of sorted discretization points. This paper introduces a bespoke Gaussian process bandit optimization method for automatically choosing these points. Our approach not only automates their one-time selection, but also dynamically adapts their positions over the course of optimization, leading to improved model learning and inference. We provide theoretical guarantees…
We present the first fully variational Bayesian inference scheme for continuous Gaussian-process-modulated Poisson processes. Such point processes are used in a variety of domains, including neuroscience, geo-statistics and astronomy, but their use is hindered by the computational cost of existing inference schemes. Our scheme: requires no discretisation of the domain; scales linearly in the number of observed events; and is many orders of magnitude faster than previous sampling based approaches. The resulting algorithm is shown to outperform standard methods on synthetic examples, coal mining…
Gaussian processes (GPs) are a good choice for function approximation as they are flexible, robust to over-fitting, and provide well-calibrated predictive uncertainty. Deep Gaussian processes (DGPs) are multi-layer generalisations of GPs, but inference in these models has proved challenging. Existing approaches to inference in DGP models assume approximate posteriors that force independence between the layers, and do not work well in practice. We present a doubly stochastic variational inference algorithm, which does not force independence between layers. With our method of inference we demons…
An important property for deep neural networks is the ability to perform robust out-of-distribution detection on previously unseen data. This property is essential for safety purposes when deploying models for real world applications. Recent studies show that probabilistic generative models can perform poorly on this task, which is surprising given that they seek to estimate the likelihood of training data. To alleviate this issue, we propose the exponentially tilted Gaussian prior distribution for the Variational Autoencoder (VAE) which pulls points onto the surface of a hyper-sphere in laten…
