Gaussian process modeling is a standard tool for building emulators for computer experiments, which are usually used to study deterministic functions, for example, a solution to a given system of partial differential equations. This work investigates applying Gaussian process modeling to a deterministic function from prediction and uncertainty quantification perspectives, where the Gaussian process model is misspecified. Specifically, we consider the case where the underlying function is fixed and from a reproducing kernel Hilbert space generated by some kernel function, and the same kernel function is used in the Gaussian process modeling as the correlation function for prediction and uncertainty quantification. While upper bounds and the optimal convergence rate of prediction in the Gaussian process modeling have been extensively studied in the literature, a comprehensive exploration of convergence rates and theoretical study of uncertainty quantification is lacking. We prove that, if one uses maximum likelihood estimation to estimate the variance in Gaussian process modeling, under different choices of the regularization parameter value, the predictor is not optimal and/or the confidence interval is not reliable. In particular, lower bounds of the prediction error under different choices of the regularization parameter value are obtained. The results indicate that, if one directly applies Gaussian process modeling to a fixed function, the reliability of the confidence interval and the optimality of the predictor cannot be achieved at the same time.
Computer experiments have become ubiquitous in science and engineering. Commonly, runs of these simulations demand considerable time and computing, making experimental design extremely important in gaining high quality information with limited time and resources. Principles of experimental design are proposed and justified which ensure high nominal, numeric, and parameter estimation accuracy for Gaussian process emulation of deterministic simulations. The space-filling properties "small fill distance" and "large separation distance" are only weakly conflicting and ensure well-controlled nomina…
University of British Columbia（不列颠哥伦比亚大学）；Yahoo! Research（雅虎研究院）
This paper analyses the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Srinivas et al., 2010) proved that the regret vanishes at the approximate rate of $O(\frac{1}{\sqrt{t}})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show t…
University of British Columbia（不列颠哥伦比亚大学）；Yahoo! Research（雅虎研究院）
This paper analyzes the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al, 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, Srinivas et al proved that the regret vanishes at the approximate rate of $O(1/\sqrt{t})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret dec…
ETH Zürich（苏黎世联邦理工学院）；Max Planck ETH Center for Learning Systems（马克斯·普朗克-苏黎世联邦理工学院学习系统中心）；Max Planck Institute for Intelligent Systems, Tübingen（图宾根马克斯·普朗克智能系统研究所）
Gaussian processes are an important regression tool with excellent analytic properties which allow for direct integration of derivative observations. However, vanilla GP methods scale cubically in the amount of observations. In this work, we propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We then prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior. To furthermore illustrate the practical applicability of our method, we then ap…
We study the problem of distributed multi-robot coverage over an unknown, nonuniform sensory field. Modeling the sensory field as a realization of a Gaussian Process and using Bayesian techniques, we devise a policy which aims to balance the tradeoff between learning the sensory function and covering the environment. We propose an adaptive coverage algorithm called Deterministic Sequencing of Learning and Coverage (DSLC) that schedules learning and coverage epochs such that its emphasis gradually shifts from exploration to exploitation while never fully ceasing to learn. Using a novel definiti…
For many expensive deterministic computer simulators, the outputs do not have replication error and the desired metamodel (or statistical emulator) is an interpolator of the observed data. Realizations of Gaussian spatial processes (GP) are commonly used to model such simulator outputs. Fitting a GP model to $n$ data points requires the computation of the inverse and determinant of $n \times n$ correlation matrices, $R$, that are sometimes computationally unstable due to near-singularity of $R$. This happens if any pair of design points are very close together in the input space. The popular a…
The maximum likelihood estimate of the correlation parameter of a Gaussian process with and without of a nugget term is studied in the case of the analysis of deterministic models.
Harvard University（哈佛大学）；University of Tokyo（东京大学）；MIT（麻省理工学院）
Generalized Chinese Remainder Theorem (CRT) has been shown to be a powerful approach to solve the ambiguity resolution problem. However, with its close relationship to number theory, study in this area is mainly from a coding theory perspective under deterministic conditions. Nevertheless, it can be proved that even with the best deterministic condition known, the probability of success in robust reconstruction degrades exponentially as the number of estimand increases. In this paper, we present the first rigorous analysis on the underlying statistical model of CRT-based multiple parameter est…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室公司）；Data61
The total variation distance is a core statistical distance between probability measures that satisfies the metric axioms, with value always falling in $[0,1]$. This distance plays a fundamental role in machine learning and signal processing: It is a member of the broader class of $f$-divergences, and it is related to the probability of error in Bayesian hypothesis testing. Since the total variation distance does not admit closed-form expressions for statistical mixtures (like Gaussian mixture models), one often has to rely in practice on costly numerical integrations or on fast Monte Carlo ap…
National University of Singapore（新加坡国立大学）；MIT（麻省理工学院）
A multi-layer deep Gaussian process (DGP) model is a hierarchical composition of GP models with a greater expressive power. Exact DGP inference is intractable, which has motivated the recent development of deterministic and stochastic approximation methods. Unfortunately, the deterministic approximation methods yield a biased posterior belief while the stochastic one is computationally costly. This paper presents an implicit posterior variational inference (IPVI) framework for DGPs that can ideally recover an unbiased posterior belief and still preserve time efficiency. Inspired by generative …
Inference in popular nonparametric Bayesian models typically relies on sampling or other approximations. This paper presents a general methodology for constructing novel tractable nonparametric Bayesian methods by applying the kernel trick to inference in a parametric Bayesian model. For example, Gaussian process regression can be derived this way from Bayesian linear regression. Despite the success of the Gaussian process framework, the kernel trick is rarely explicitly considered in the Bayesian literature. In this paper, we aim to fill this gap and demonstrate the potential of applying the …
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, …
An Efficient Spatial Branch-and-Bound Algorithm for Global Optimization of Gaussian Process Posterior Mean Functions
Department of Chemical and Biological Engineering, University of Wisconsin-Madison（威斯康星大学麦迪逊分校化学与生物工程系）；Department of Chemical and Biomolecular Engineering, The Ohio State University（俄亥俄州立大学化学与生物分子工程系）；Department of Computing, Imperial College London（伦敦帝国理工学院计算系）
Despite the ubiquity of the Gaussian process regression model, few theoretical results are available that account for the fact that parameters of the covariance kernel typically need to be estimated from the dataset. This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless dataset. Specifically, we consider the scenario where the scale parameter of a Sobolev kernel (such as a Matérn kernel) is estimated by maximum likelihood. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptat…
MIT（麻省理工学院）；University of Chicago（芝加哥大学）；UBC（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）；TTI-Chicago（芝加哥丰田技术研究所）
We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, ob…
