高斯过程（Kriging）是插值型数据驱动模型，广泛应用于多个学科。通常，高斯过程在数据集上训练后被嵌入优化问题中作为代理模型。这些优化问题是非凸的，需要进行全局优化。然而，以往文献观察到计算负担限制了确定性全局优化仅适用于在少量数据点上训练的高斯过程。我们提出了一种嵌入已训练高斯过程的确定性全局优化的降维空间公式。在优化过程中，分支定界求解器仅在自由度上进行分支，McCormick松弛通过显式高斯过程模型进行传播。该方法还显著减小了用于下界和上界的子问题规模并降低了计算成本。为进一步加速收敛，我们推导了高斯过程常用协方差函数的包络，以及贝叶斯优化中使用的采集函数的紧松弛，包括期望改进（expected improvement）、改进概率（probability of improvement）和下置信界（lower confidence bound）。总体而言，与最先进的方法相比，我们将计算时间减少了几个数量级，从而克服了以往的计算负担。我们展示了所提方法的性能和可扩展性，并将其应用于采集函数全局优化的贝叶斯优化和机会约束规划。高斯过程模型、采集函数和训练脚本在“MeLOn - Machine Learning Models for Optimization”工具箱（https://git.rwth-aachen.de/avt.svt/public/MeLOn）中开源提供。
英文摘要：
Gaussian processes~(Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the degrees of freedom and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the "MeLOn - Machine Learning Models for Optimization" toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn).
Aalto university（阿尔托大学）；Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））
We propose deep convolutional Gaussian processes, a deep Gaussian process architecture with convolutional structure. The model is a principled Bayesian framework for detecting hierarchical combinations of local features for image classification. We demonstrate greatly improved image classification performance compared to current Gaussian process approaches on the MNIST and CIFAR-10 datasets. In particular, we improve CIFAR-10 accuracy by over 10 percentage points.
Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for broadening the applicability of Gaussian processes. This is done in two ways. Firstly, we develop pathwise conditioning techniques for Gaussian processes, which allow one to express posterior random functions as prior random functions plus a dependent update term. We introduce a wide class of efficient approximations built from this viewpoint, which can be randomly …
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）
Gaussian processes are one of the dominant approaches in Bayesian learning. Although the approach has been applied to numerous problems with great success, it has a few fundamental limitations. Multiple methods in literature have addressed these limitations. However, there has not been a comprehensive survey of the topics as of yet. Most existing surveys focus on only one particular variant of Gaussian processes and their derivatives. This survey details the core motivations for using Gaussian processes, their mathematical formulations, limitations, and research themes that have flourished ove…
The Alan Turing Institute（艾伦·图灵研究所）；University of Warwick（华威大学）
We present a multi-task learning formulation for Deep Gaussian processes (DGPs), through non-linear mixtures of latent processes. The latent space is composed of private processes that capture within-task information and shared processes that capture across-task dependencies. We propose two different methods for segmenting the latent space: through hard coding shared and task-specific processes or through soft sharing with Automatic Relevance Determination kernels. We show that our formulation is able to improve the learning performance and transfer information between the tasks, outperforming…
We introduce stochastic variational inference for Gaussian process models. This enables the application of Gaussian process (GP) models to data sets containing millions of data points. We show how GPs can be vari- ationally decomposed to depend on a set of globally relevant inducing variables which factorize the model in the necessary manner to perform variational inference. Our ap- proach is readily extended to models with non-Gaussian likelihoods and latent variable models based around Gaussian processes. We demonstrate the approach on a simple toy problem and two real world data sets.
Motivated by objects such as electric fields or fluid streams, we study the problem of learning stochastic fields, i.e. stochastic processes whose samples are fields like those occurring in physics and engineering. Considering general transformations such as rotations and reflections, we show that spatial invariance of stochastic fields requires an inference model to be equivariant. Leveraging recent advances from the equivariance literature, we study equivariance in two classes of models. Firstly, we fully characterise equivariant Gaussian processes. Secondly, we introduce Steerable Condition…
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.
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…
Ulsan National Institute of Science and Technology（蔚山科学技术院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；INEEJI
Recent advances in Deep Gaussian Processes (DGPs) show the potential to have more expressive representation than that of traditional Gaussian Processes (GPs). However, there exists a pathology of deep Gaussian processes that their learning capacities reduce significantly when the number of layers increases. In this paper, we present a new analysis in DGPs by studying its corresponding nonlinear dynamic systems to explain the issue. Existing work reports the pathology for the squared exponential kernel function. We extend our investigation to four types of common stationary kernel functions. Th…
St. Petersburg State University（圣彼得堡国立大学）；Imperial College London（帝国理工学院伦敦分校）；Centre for Artificial Intelligence, University College London（伦敦大学学院人工智能中心）；Secondmind（Secondmind公司）；St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院圣彼得堡斯捷克洛夫数学研究所彼得堡分部）
We construct flexible likelihoods for multi-output Gaussian process models that leverage neural networks as components. We make use of sparse variational inference methods to enable scalable approximate inference for the resulting class of models. An attractive feature of these models is that they can admit analytic predictive means even when the likelihood is non-linear and the predictive distributions are non-Gaussian. We validate the modeling potential of these models in a variety of experiments in both the supervised and unsupervised setting. We demonstrate that the flexibility of these `n…
University of Oxford（牛津大学）；University College London（伦敦大学学院）；University of Cambridge（剑桥大学）；Imperial College London（帝国理工学院）；St. Petersburg State University（圣彼得堡国立大学）
Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas …
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…
Université de Lille（里尔大学）；CNRS/CRIStAL（法国国家科学研究中心/CRIStAL实验室）；Inria Lille（法国里尔国家信息与自动化研究所）
Bayesian Optimization using Gaussian Processes is a popular approach to deal with the optimization of expensive black-box functions. However, because of the a priori on the stationarity of the covariance matrix of classic Gaussian Processes, this method may not be adapted for non-stationary functions involved in the optimization problem. To overcome this issue, a new Bayesian Optimization approach is proposed. It is based on Deep Gaussian Processes as surrogate models instead of classic Gaussian Processes. This modeling technique increases the power of representation to capture the non-station…
As a generalization of the work in [Lee et al., 2017], this note briefly discusses when the prior of a neural network output follows a Gaussian process, and how a neural-network-induced Gaussian process is formulated. The posterior mean functions of such a Gaussian process regression lie in the reproducing kernel Hilbert space defined by the neural-network-induced kernel. In the case of two-layer neural networks, the induced Gaussian processes provide an interpretation of the reproducing kernel Hilbert spaces whose union forms a Barron space.
Gaussian processes are used in machine learning to learn input-output mappings from observed data. Gaussian process regression is based on imposing a Gaussian process prior on the unknown regressor function and statistically conditioning it on the observed data. In system identification, Gaussian processes are used to form time series prediction models such as non-linear finite-impulse response (NFIR) models as well as non-linear autoregressive (NARX) models. Gaussian process state-space models (GPSS) can be used to learn the dynamic and measurement models for a state-space representation of t…
Gaussian process priors are commonly used in aerospace design for performing Bayesian optimization. Nonetheless, Gaussian processes suffer two significant drawbacks: outliers are a priori assumed unlikely, and the posterior variance conditioned on observed data depends only on the locations of those data, not the associated sample values. Student's-T processes are a generalization of Gaussian processes, founded on the Student's-T distribution instead of the Gaussian distribution. Student's-T processes maintain the primary advantages of Gaussian processes (kernel function, analytic update rule)…
Australian Centre for Field Robotics, The University of Sydney（悉尼大学澳大利亚现场机器人中心）
This paper evaluates heterogeneous information fusion using multi-task Gaussian processes in the context of geological resource modeling. Specifically, it empirically demonstrates that information integration across heterogeneous information sources leads to superior estimates of all the quantities being modeled, compared to modeling them individually. Multi-task Gaussian processes provide a powerful approach for simultaneous modeling of multiple quantities of interest while taking correlations between these quantities into consideration. Experiments are performed on large scale real sensor da…
We propose a novel Bayesian nonparametric method to learn translation-invariant relationships on non-Euclidean domains. The resulting graph convolutional Gaussian processes can be applied to problems in machine learning for which the input observations are functions with domains on general graphs. The structure of these models allows for high dimensional inputs while retaining expressibility, as is the case with convolutional neural networks. We present applications of graph convolutional Gaussian processes to images and triangular meshes, demonstrating their versatility and effectiveness, com…
Gaussian processes are ubiquitous in nature and engineering. A case in point is a class of neural networks in the infinite-width limit, whose priors correspond to Gaussian processes. Here we perturbatively extend this correspondence to finite-width neural networks, yielding non-Gaussian processes as priors. The methodology developed herein allows us to track the flow of preactivation distributions by progressively integrating out random variables from lower to higher layers, reminiscent of renormalization-group flow. We further develop a perturbative procedure to perform Bayesian inference wit…
Graph Gaussian Processes (GGPs) provide a data-efficient solution on graph structured domains. Existing approaches have focused on static structures, whereas many real graph data represent a dynamic structure, limiting the applications of GGPs. To overcome this we propose evolving-Graph Gaussian Processes (e-GGPs). The proposed method is capable of learning the transition function of graph vertices over time with a neighbourhood kernel to model the connectivity and interaction changes between vertices. We assess the performance of our method on time-series regression problems where graphs evol…
RISE Research Institutes of Sweden AB（瑞典RISE研究院）；Elekta（医科达）
We present the elliptical processes -- a family of non-parametric probabilistic models that subsumes the Gaussian process and the Student-t process. This generalization includes a range of new fat-tailed behaviors yet retains computational tractability. We base the elliptical processes on a representation of elliptical distributions as a continuous mixture of Gaussian distributions and derive closed-form expressions for the marginal and conditional distributions. We perform numerical experiments on robust regression using an elliptical process defined by a piecewise constant mixing distributio…
We propose a simple method that combines neural networks and Gaussian processes. The proposed method can estimate the uncertainty of outputs and flexibly adjust target functions where training data exist, which are advantages of Gaussian processes. The proposed method can also achieve high generalization performance for unseen input configurations, which is an advantage of neural networks. With the proposed method, neural networks are used for the mean functions of Gaussian processes. We present a scalable stochastic inference procedure, where sparse Gaussian processes are inferred by stochast…
We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Studen…
Real-world measurement noise in applications like robotics is often correlated in time, but we typically assume i.i.d. Gaussian noise for filtering. We propose general Gaussian Processes as a non-parametric model for correlated measurement noise that is flexible enough to accurately reflect correlation in time, yet simple enough to enable efficient computation. We show that this model accurately reflects the measurement noise resulting from vision-based Simultaneous Localization and Mapping (SLAM), and argue that it provides a flexible means of modeling measurement noise for a wide variety of …
We provide the strong approximation of empirical copula processes by a Gaussian process. In addition we establish a strong approximation of the smoothed empirical copula processes and a law of iterated logarithm.
In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easil…
A gentle introduction to Gaussian processes (GPs). The three parts of the document consider GPs for regression, classification, and dimensionality reduction.
In this paper, we study central and non-central limit theorems for partial sum of functionals of general stationary Gaussian fields. We apply our result to study drift parameter estimation problems for some stochastic differential equations related to stationary Gaussian processes.
This article gives a new proof that fully connected neural networks with random weights and biases converge to Gaussian processes in the regime where the input dimension, output dimension, and depth are kept fixed, while the hidden layer widths tend to infinity. Unlike prior work, convergence is shown assuming only moment conditions for the distribution of weights and for quite general non-linearities.
We derive expressions for the predicitive information rate (PIR) for the class of autoregressive Gaussian processes AR(N), both in terms of the prediction coefficients and in terms of the power spectral density. The latter result suggests a duality between the PIR and the multi-information rate for processes with mutually inverse power spectra (i.e. with poles and zeros of the transfer function exchanged). We investigate the behaviour of the PIR in relation to the multi-information rate for some simple examples, which suggest, somewhat counter-intuitively, that the PIR is maximised for very `s…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the other to gradient matching. All methods have greater accuracy than previously suggested Gaussian Process approaches. We also suggest a general approach that can yield error estimates from any standard ODE solver.
The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of non-Gaussian processes obtained by regular non-linear transformations of Gaussian processes. We provide the increasing-domain asymptotic properties of the (Gaussian) maximum likelihood and cross validation estimators of the covariance parameters of a non-Gaussian process of this class. We show that these estimators are consistent and asymptotically normal, alt…
Deep Learning Gaussian Processes (DL-GP) are proposed as a methodology for analyzing (approximating) computer models that produce heteroskedastic and high-dimensional output. Computer simulation models have many areas of applications, including social-economic processes, agriculture, environmental, biology, engineering and physics problems. A deterministic transformation of inputs is performed by deep learning and predictions are calculated by traditional Gaussian Processes. We illustrate our methodology using a simulation of motorcycle accidents and simulations of an Ebola outbreak. Finally, …
There are all kinds of weak dependence. For example, strong mixing. Short-range dependence (SRD) is also a form of weak dependence. It occurs in the context of processes that are subordinated to the Gaussian. Is a SRD process strong mixing if the underlying Gaussian process is long-range dependent? We show that this is not necessarily the case.
In this paper we derive Piterbarg's max-discretisation theorem for two different grids considering centered stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over $[0,T]$ and over a grid $ \mathfrak{R}(δ_1(T))=\{kδ_1(T): k=0,1,\cdots\}$. In this paper we extend recent findings by considering additionally the \bE{maximum} over another grid $ \mathfrak{R}(δ_2(T))$. We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choice…
We present a non-parametric joint estimation method for fMRI task activation values and the hemodynamic response function (HRF). The HRF is modeled as a Gaussian process, making continuous evaluation possible for jittered paradigms and providing a variance estimate at each point.
Various approaches to stochastic processes exist, noting that key properties such as measurability and continuity are not trivially satisfied. We introduce a new theory for Gaussian processes using improper linear functionals. Using a collection of i.i.d. standard normal variables, we define Gaussian white noise and discuss its properties. This is extended to general Gaussian processes on Hilbert space, where the variance is allowed to be any suitable operator. Our main focus is $L^2$ spaces, and we discuss criteria for Gaussian processes to be continuous in this setting. Finally, we outline a…
Gaussian processes occupy one of the leading places in modern statistics and probability theory due to their importance and a wealth of strong results. The common use of Gaussian processes is in connection with problems related to estimation, detection, and many statistical or machine learning models. With the fast development of Gaussian process applications, it is necessary to consolidate the fundamentals of vector-valued stochastic processes, in particular multivariate Gaussian processes, which is the essential theory for many applied problems with multiple correlated responses. In this pap…
In this paper, we study the quantitative behavior of a spatial risk measure corresponding to a damage function and a region, taking into account the spatial dependence of the underlying process. This kind of risk measure has already been introduced and studied for some max-stable processes in [Koch2015]. In this paper, we consider isotropic Gaussian processes and the excess damage function over a threshold. We performed a simulation study and a real data study.
We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.
In this paper, we derive tail approximations of integrals of exponential functions of Gaussian random fields with varying mean functions and approximations of the associated point processes. This study is motivated naturally by multiple applications such as hypothesis testing for spatial models, study of the distribution of Bayesian marginal likelihood and Bayes factor, and financial applications.
The log-Gaussian Cox process is a flexible and popular class of point pattern models for capturing spatial and space-time dependence for point patterns. Model fitting requires approximation of stochastic integrals which is implemented through discretization over the domain of interest. With fine scale discretization, inference based on Markov chain Monte Carlo is computationally burdensome because of the cost of matrix decompositions and storage, such as the Cholesky, for high dimensional covariance matrices associated with latent Gaussian variables. This article addresses these computational …
Spatio-temporal processes in environmental applications are often assumed to follow a Gaussian model, possibly after some transformation. However, heterogeneity in space and time might have a pattern that will not be accommodated by transforming the data. In this scenario, modelling the variance laws is an appealing alternative. This work adds flexibility to the usual Multivariate Dynamic Gaussian model by defining the process as a scale mixture between a Gaussian and log-Gaussian processes. The scale is represented by a process varying smoothly over space and time which is allowed to depend o…
We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].
Recent research has shown the potential utility of Deep Gaussian Processes. These deep structures are probability distributions, designed through hierarchical construction, which are conditionally Gaussian. In this paper, the current published body of work is placed in a common framework and, through recursion, several classes of deep Gaussian processes are defined. The resulting samples generated from a deep Gaussian process have a Markovian structure with respect to the depth parameter, and the effective depth of the resulting process is interpreted in terms of the ergodicity, or non-ergodic…
We study rates of convergence in central limit theorems for partial sum of functionals of general stationary and non-stationary Gaussian sequences, using optimal tools from analysis on Wiener space. We apply our result to study drift parameter estimation problems for some stochastic differential equations driven by fractional Brownian motion with fixed-time-step observations.
Generalized Gaussian processes (GGPs) are highly flexible models that combine latent GPs with potentially non-Gaussian likelihoods from the exponential family. GGPs can be used in a variety of settings, including GP classification, nonparametric count regression, modeling non-Gaussian spatial data, and analyzing point patterns. However, inference for GGPs can be analytically intractable, and large datasets pose computational challenges due to the inversion of the GP covariance matrix. We propose a Vecchia-Laplace approximation for GGPs, which combines a Laplace approximation to the non-Gaussia…
We consider the optimal transport problem between multivariate Gaussian stationary stochastic processes. The transportation effort is the variance of the filtered discrepancy process. The main contribution of this technical note is to show that the corresponding solution leads to a weighted Hellinger distance between multivariate power spectral densities. Then, we propose a spectral estimation approach in the case of indirect measurements which is based on this distance.
