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, we propose a scalable variational inference algorithm for GPRN, which not only captures the abundant posterior dependencies but also is much more efficient for massive outputs. We tensorize the output space and introduce tensor/matrix-normal variational posteriors to capture the posterior correlations and to reduce the parameters. We jointly optimize all the parameters and exploit the inherent Kronecker product structure in the variational model evidence lower bound to accelerate the computation. We demonstrate the advantages of our method in several real-world applications.
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.
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.
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 …
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…
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…
While unsupervised variational autoencoders (VAE) have become a powerful tool in neuroimage analysis, their application to supervised learning is under-explored. We aim to close this gap by proposing a unified probabilistic model for learning the latent space of imaging data and performing supervised regression. Based on recent advances in learning disentangled representations, the novel generative process explicitly models the conditional distribution of latent representations with respect to the regression target variable. Performing a variational inference procedure on this model leads to j…
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…
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…
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 …
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.
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…
Learning-enabled components (LECs) are widely used in cyber-physical systems (CPS) since they can handle the uncertainty and variability of the environment and increase the level of autonomy. However, it has been shown that LECs such as deep neural networks (DNN) are not robust and adversarial examples can cause the model to make a false prediction. The paper considers the problem of efficiently detecting adversarial examples in LECs used for regression in CPS. The proposed approach is based on inductive conformal prediction and uses a regression model based on variational autoencoder. The arc…
We implement gradient-based variational inference routines for Wishart and inverse Wishart processes, which we apply as Bayesian models for the dynamic, heteroskedastic covariance matrix of a multivariate time series. The Wishart and inverse Wishart processes are constructed from i.i.d. Gaussian processes, existing variational inference algorithms for which form the basis of our approach. These methods are easy to implement as a black-box and scale favorably with the length of the time series, however, they fail in the case of the Wishart process, an issue we resolve with a simple modification…
We develop a novel probabilistic generative model based on the variational autoencoder approach. Notable aspects of our architecture are: a novel way of specifying the latent variables prior, and the introduction of an ordinality enforcing unit. We describe how to do supervised, unsupervised and semi-supervised learning, and nominal and ordinal classification, with the model. We analyze generative properties of the approach, and the classification effectiveness under nominal and ordinal classification, using two benchmark datasets. Our results show that our model can achieve comparable results…
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…
Seoul National University（首尔国立大学）；Imperial College London（帝国理工学院）
This paper proposes a new high dimensional regression method by merging Gaussian process regression into a variational autoencoder framework. In contrast to other regression methods, the proposed method focuses on the case where output responses are on a complex high dimensional manifold, such as images. Our contributions are summarized as follows: (i) A new regression method estimating high dimensional image responses, which is not handled by existing regression algorithms, is proposed. (ii) The proposed regression method introduces a strategy to learn the latent space as well as the encoder …
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…
King Digital Entertainment, Activision Blizzard Group（动视暴雪集团King数字娱乐公司）；Uppsala University（乌普萨拉大学）
Deep clustering (DC) has become the state-of-the-art for unsupervised clustering. In principle, DC represents a variety of unsupervised methods that jointly learn the underlying clusters and the latent representation directly from unstructured datasets. However, DC methods are generally poorly applied due to high operational costs, low scalability, and unstable results. In this paper, we first evaluate several popular DC variants in the context of industrial applicability using eight empirical criteria. We then choose to focus on variational deep clustering (VDC) methods, since they mostly mee…
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…
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…
As a non-parametric Bayesian model which produces informative predictive distribution, Gaussian process (GP) has been widely used in various fields, like regression, classification and optimization. The cubic complexity of standard GP however leads to poor scalability, which poses challenges in the era of big data. Hence, various scalable GPs have been developed in the literature in order to improve the scalability while retaining desirable prediction accuracy. This paper devotes to investigating the methodological characteristics and performance of representative global and local scalable GPs…
Online Passive-Aggressive (PA) learning is a class of online margin-based algorithms suitable for a wide range of real-time prediction tasks, including classification and regression. PA algorithms are formulated in terms of deterministic point-estimation problems governed by a set of user-defined hyperparameters: the approach fails to capture model/prediction uncertainty and makes their performance highly sensitive to hyperparameter configurations. In this paper, we introduce a novel PA learning framework for regression that overcomes the above limitations. We contribute a Bayesian state-space…
Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization. Typically, BO is powered by a Gaussian process (GP), whose algorithmic complexity is cubic in the number of evaluations. Hence, GP-based BO cannot leverage large amounts of past or related function evaluations, for example, to warm start the BO procedure. We develop a multiple adaptive Bayesian linear regression model as a scalable alternative whose complexity is linear in the number of observations. The multiple Bayesian linear regression models are coupled through a shared feedforward neur…
The Cut posterior and related Semi-Modular Inference are Generalised Bayes methods for Modular Bayesian evidence combination. Analysis is broken up over modular sub-models of the joint posterior distribution. Model-misspecification in multi-modular models can be hard to fix by model elaboration alone and the Cut posterior and SMI offer a way round this. Information entering the analysis from misspecified modules is controlled by an influence parameter $η$ related to the learning rate. This paper contains two substantial new methods. First, we give variational methods for approximating the Cut …
One major drawback of state-of-the-art artificial intelligence is its lack of explainability. One approach to solve the problem is taking causality into account. Causal mechanisms can be described by structural causal models. In this work, we propose a method for estimating bivariate structural causal models using a combination of normalising flows applied to density estimation and variational Gaussian process regression for post-nonlinear models. It facilitates causal discovery, i.e. distinguishing cause and effect, by either the independence of cause and residual or a likelihood ratio test. …
University of Edinburgh（爱丁堡大学）；MIT–IBM Watson AI Lab（麻省理工-IBM沃森人工智能实验室）；University of Oslo（奥斯陆大学）
Determining the positions of neurons in an extracellular recording is useful for investigating functional properties of the underlying neural circuitry. In this work, we present a Bayesian modelling approach for localizing the source of individual spikes on high-density, microelectrode arrays. To allow for scalable inference, we implement our model as a variational autoencoder and perform amortized variational inference. We evaluate our method on both biophysically realistic simulated and real extracellular datasets, demonstrating that it is more accurate than and can improve spike sorting per…
Stanford University（斯坦福大学）；Facebook AI Research（Facebook人工智能研究院）；University of California, Los Angeles（加州大学洛杉矶分校）
A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizi…
Imperial College London（帝国理工学院）；Bogazici University（海峡大学）
We present the development of a semi-supervised regression method using variational autoencoders (VAE), which is customized for use in soft sensing applications. We motivate the use of semi-supervised learning considering the fact that process quality variables are not collected at the same frequency as other process variables leading to many unlabelled records in operational datasets. These unlabelled records are not possible to use for training quality variable predictions based on supervised learning methods. Use of VAEs for unsupervised learning is well established and recently they were u…
We develop an exact and scalable algorithm for one-dimensional Gaussian process regression with Matérn correlations whose smoothness parameter $ν$ is a half-integer. The proposed algorithm only requires $\mathcal{O}(ν^3 n)$ operations and $\mathcal{O}(νn)$ storage. This leads to a linear-cost solver since $ν$ is chosen to be fixed and usually very small in most applications. The proposed method can be applied to multi-dimensional problems if a full grid or a sparse grid design is used. The proposed method is based on a novel theory for Matérn correlation functions. We find that a suitable rear…
Gaussian process (GP) regression is a flexible, nonparametric approach to regression that naturally quantifies uncertainty. In many applications, the number of responses and covariates are both large, and a goal is to select covariates that are related to the response. For this setting, we propose a novel, scalable algorithm, coined VGPR, which optimizes a penalized GP log-likelihood based on the Vecchia GP approximation, an ordered conditional approximation from spatial statistics that implies a sparse Cholesky factor of the precision matrix. We traverse the regularization path from strong to…
Cyber-physical systems (CPSs) use learning-enabled components (LECs) extensively to cope with various complex tasks under high-uncertainty environments. However, the dataset shifts between the training and testing phase may lead the LECs to become ineffective to make large-error predictions, and further, compromise the safety of the overall system. In our paper, we first provide the formal definitions for different types of dataset shifts in learning-enabled CPS. Then, we propose an approach to detect the dataset shifts effectively for regression problems. Our approach is based on the inductiv…
In this tutorial we explain the inference procedures developed for the sparse Gaussian process (GP) regression and Gaussian process latent variable model (GPLVM). Due to page limit the derivation given in Titsias (2009) and Titsias & Lawrence (2010) is brief, hence getting a full picture of it requires collecting results from several different sources and a substantial amount of algebra to fill-in the gaps. Our main goal is thus to collect all the results and full derivations into one place to help speed up understanding this work. In doing so we present a re-parametrisation of the inference t…
Collected data, which is used for analysis or prediction tasks, often have a hierarchical structure, for example, data from various people performing the same task. Modeling the data's structure can improve the reliability of the derived results and prediction performance of newly unobserved data. Bayesian modeling provides a tool-kit for designing hierarchical models. However, Markov Chain Monte Carlo methods which are commonly used for parameter estimation are computationally expensive. This often renders its use for many applications not applicable. However, variational Bayesian methods all…
Deep generative models have achieved great success in unsupervised learning with the ability to capture complex nonlinear relationships between latent generating factors and observations. Among them, a factorized hierarchical variational autoencoder (FHVAE) is a variational inference-based model that formulates a hierarchical generative process for sequential data. Specifically, an FHVAE model can learn disentangled and interpretable representations, which have been proven useful for numerous speech applications, such as speaker verification, robust speech recognition, and voice conversion. Ho…
Erasmus University Rotterdam（鹿特丹伊拉斯姆斯大学）；Stanford University（斯坦福大学）
We propose a general algorithm for approximating nonstandard Bayesian posterior distributions. The algorithm minimizes the Kullback-Leibler divergence of an approximating distribution to the intractable posterior distribution. Our method can be used to approximate any posterior distribution, provided that it is given in closed form up to the proportionality constant. The approximation can be any distribution in the exponential family or any mixture of such distributions, which means that it can be made arbitrarily precise. Several examples illustrate the speed and accuracy of our approximation…
As Deep Learning continues to yield successful applications in Computer Vision, the ability to quantify all forms of uncertainty is a paramount requirement for its safe and reliable deployment in the real-world. In this work, we leverage the formulation of variational inference in function space, where we associate Gaussian Processes (GPs) to both Bayesian CNN priors and variational family. Since GPs are fully determined by their mean and covariance functions, we are able to obtain predictive uncertainty estimates at the cost of a single forward pass through any chosen CNN architecture and for…
In the U.S., corn is the most produced crop and has been an essential part of the American diet. To meet the demand for supply chain management and regional food security, accurate and timely large-scale corn yield prediction is attracting more attention in precision agriculture. Recently, remote sensing technology and machine learning methods have been widely explored for crop yield prediction. Currently, most county-level yield prediction models use county-level mean variables for prediction, ignoring much detailed information. Moreover, inconsistent spatial resolution between crop area and …
We recently proposed a general algorithm for approximating nonstandard Bayesian posterior distributions by minimization of their Kullback-Leibler divergence with respect to a more convenient approximating distribution. In this note we offer details on how to efficiently implement this algorithm in practice. We also suggest default choices for the form of the posterior approximation, the number of iterations, the step size, and other user choices. By using these defaults it becomes possible to construct good posterior approximations for hierarchical models completely automatically.
We develop a fully automatic Bayesian Lasso via variational inference. This is a scalable procedure for approximating the posterior distribution. Special attention is driven to the knot selection in regression spline. In order to carry through our proposal, a full automatic variational Bayesian Lasso, a Jefferey's prior is proposed for the hyperparameters and a decision theoretical approach is introduced to decide if a knot is selected or not. Extensive simulation studies were developed to ensure the effectiveness of the proposed algorithms. The performance of the algorithms were also tested i…
The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either r…
