Understanding and Comparing Scalable Gaussian Process Regression for Big Data

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 including sparse approximations and local aggregations from four main perspectives: scalability, capability, controllability and robustness. The numerical experiments on two toy examples and five real-world datasets with up to 250K points offer the following findings. In terms of scalability, most of the scalable GPs own a time complexity that is linear to the training size. In terms of capability, the sparse approximations capture the long-term spatial correlations, the local aggregations capture the local patterns but suffer from over-fitting in some scenarios. In terms of controllability, we could improve the performance of sparse approximations by simply increasing the inducing size. But this is not the case for local aggregations. In terms of robustness, local aggregations are robust to various initializations of hyperparameters due to the local attention mechanism. Finally, we highlight that the proper hybrid of global and local scalable GPs may be a promising way to improve both the model capability and scalability for big data.Comment: 25 pages, 15 figures, preprint submitted to KB

Sequential Design for Gaussian Process Surrogates in Noisy Level Set Estimation

We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels and sequential design frameworks. Our focus is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. We introduce the use of four GP-based metamodels in level set estimation that are robust to noise misspecification, and evaluate the performance of them. In conjunction with these metamodels, we develop several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions for the proposed metamodels. To expedite sequential design in stochastic experiments, we also develop adaptive batching designs, which are natural extensions of sequential design heuristics with the benefit of replication growing as response features are learned, inputs concentrate, and the metamodeling overhead rises. We develop four novel schemes that simultaneously or sequentially determine the sequential design inputs and the respective number of replicates. Our schemes are benchmarked by using synthetic examples and an application in quantitative finance (Bermudan option pricing)

hetGP: Heteroskedastic Gaussian Process Modeling and Sequential Design in R

An increasing number of time-consuming simulators exhibit a complex noise structure that depends on the inputs. For conducting studies with limited budgets of evaluations, new surrogate methods are required in order to simultaneously model the mean and variance fields. To this end, we present the hetGP package, implementing many recent advances in Gaussian process modeling with input-dependent noise. First, we describe a simple, yet efficient, joint modeling framework that relies on replication for both speed and accuracy. Then we tackle the issue of data acquisition leveraging replication and exploration in a sequential manner for various goals, such as for obtaining a globally accurate model, for optimization, or for contour finding. Reproducible illustrations are provided throughout

Achieving robustness to aleatoric uncertainty with heteroscedastic Bayesian optimisation

Bayesian optimisation is a sample-efficient search methodology that holds great promise for accelerating drug and materials discovery programs. A frequently-overlooked modelling consideration in Bayesian optimisation strategies however, is the representation of heteroscedastic aleatoric uncertainty. In many practical applications it is desirable to identify inputs with low aleatoric noise, an example of which might be a material composition which consistently displays robust properties in response to a noisy fabrication process. In this paper, we propose a heteroscedastic Bayesian optimisation scheme capable of representing and minimising aleatoric noise across the input space. Our scheme employs a heteroscedastic Gaussian process (GP) surrogate model in conjunction with two straightforward adaptations of existing acquisition functions. First, we extend the augmented expected improvement (AEI) heuristic to the heteroscedastic setting and second, we introduce the aleatoric noise-penalised expected improvement (ANPEI) heuristic. Both methodologies are capable of penalising aleatoric noise in the suggestions and yield improved performance relative to homoscedastic Bayesian optimisation and random sampling on toy problems as well as on two real-world scientific datasets. Code is available at: \url{https://github.com/Ryan-Rhys/Heteroscedastic-BO

Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial

On top of machine learning models, uncertainty quantification (UQ) functions as an essential layer of safety assurance that could lead to more principled decision making by enabling sound risk assessment and management. The safety and reliability improvement of ML models empowered by UQ has the potential to significantly facilitate the broad adoption of ML solutions in high-stakes decision settings, such as healthcare, manufacturing, and aviation, to name a few. In this tutorial, we aim to provide a holistic lens on emerging UQ methods for ML models with a particular focus on neural networks and the applications of these UQ methods in tackling engineering design as well as prognostics and health management problems. Toward this goal, we start with a comprehensive classification of uncertainty types, sources, and causes pertaining to UQ of ML models. Next, we provide a tutorial-style description of several state-of-the-art UQ methods: Gaussian process regression, Bayesian neural network, neural network ensemble, and deterministic UQ methods focusing on spectral-normalized neural Gaussian process. Established upon the mathematical formulations, we subsequently examine the soundness of these UQ methods quantitatively and qualitatively (by a toy regression example) to examine their strengths and shortcomings from different dimensions. Then, we review quantitative metrics commonly used to assess the quality of predictive uncertainty in classification and regression problems. Afterward, we discuss the increasingly important role of UQ of ML models in solving challenging problems in engineering design and health prognostics. Two case studies with source codes available on GitHub are used to demonstrate these UQ methods and compare their performance in the life prediction of lithium-ion batteries at the early stage and the remaining useful life prediction of turbofan engines