Author name cluster

Carl Edward Rasmussen

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

19 papers

1 author row

TMLR Journal 2024 Journal Article

Integrated Variational Fourier Features for Fast Spatial Modelling with Gaussian Processes

Talay M Cheema
Carl Edward Rasmussen

Sparse variational approximations are popular methods for scaling up inference and learning in Gaussian processes to larger datasets. For $N$ training points, exact inference has $O(N^3)$ cost; with $M \ll N$ features, state of the art sparse variational methods have $O(NM^2)$ cost. Recently, methods have been proposed using more sophisticated features; these promise $O(M^3)$ cost, with good performance in low dimensional tasks such as spatial modelling, but they only work with a very limited class of kernels, excluding some of the most commonly used. In this work, we propose integrated Fourier features, which extends these performance benefits to a very broad class of stationary covariance functions. We motivate the method and choice of parameters from a convergence analysis and empirical exploration, and show practical speedup in synthetic and real world spatial regression tasks.

JMLR Journal 2024 Journal Article

Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees

Alexander Terenin
David R. Burt
Artem Artemev
Seth Flaxman
Mark van der Wilk
Carl Edward Rasmussen
Hong Ge

Gaussian processes are frequently deployed as part of larger machine learning and decision-making systems, for instance in geospatial modeling, Bayesian optimization, or in latent Gaussian models. Within a system, the Gaussian process model needs to perform in a stable and reliable manner to ensure it interacts correctly with other parts of the system. In this work, we study the numerical stability of scalable sparse approximations based on inducing points. To do so, we first review numerical stability, and illustrate typical situations in which Gaussian process models can be unstable. Building on stability theory originally developed in the interpolation literature, we derive sufficient and in certain cases necessary conditions on the inducing points for the computations performed to be numerically stable. For low-dimensional tasks such as geospatial modeling, we propose an automated method for computing inducing points satisfying these conditions. This is done via a modification of the cover tree data structure, which is of independent interest. We additionally propose an alternative sparse approximation for regression with a Gaussian likelihood which trades off a small amount of performance to further improve stability. We provide illustrative examples showing the relationship between stability of calculations and predictive performance of inducing point methods on spatial tasks. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2024. ( edit, beta )

NeurIPS Conference 2022 Conference Paper

Sparse Gaussian Process Hyperparameters: Optimize or Integrate?

Vidhi Lalchand
Wessel Bruinsma
David Burt
Carl Edward Rasmussen

The kernel function and its hyperparameters are the central model selection choice in a Gaussian process (Rasmussen and Williams, 2006). Typically, the hyperparameters of the kernel are chosen by maximising the marginal likelihood, an approach known as Type-II maximum likelihood (ML-II). However, ML-II does not account for hyperparameter uncertainty, and it is well-known that this can lead to severely biased estimates and an underestimation of predictive uncertainty. While there are several works which employ fully Bayesian characterisation of GPs, relatively few propose such approaches for the sparse GPs paradigm. In this work we propose an algorithm for sparse Gaussian process regression which leverages MCMC to sample from the hyperparameter posterior within the variational inducing point framework of (Titsias, 2009). This work is closely related to (Hensman et al, 2015b) but side-steps the need to sample the inducing points, thereby significantly improving sampling efficiency in the Gaussian likelihood case. We compare this scheme against natural baselines in literature along with stochastic variational GPs (SVGPs) along with an extensive computational analysis.

NeurIPS Conference 2021 Conference Paper

Kernel Identification Through Transformers

Fergus Simpson
Ian Davies
Vidhi Lalchand
Alessandro Vullo
Nicolas Durrande
Carl Edward Rasmussen

Kernel selection plays a central role in determining the performance of Gaussian Process (GP) models, as the chosen kernel determines both the inductive biases and prior support of functions under the GP prior. This work addresses the challenge of constructing custom kernel functions for high-dimensional GP regression models. Drawing inspiration from recent progress in deep learning, we introduce a novel approach named KITT: Kernel Identification Through Transformers. KITT exploits a transformer-based architecture to generate kernel recommendations in under 0. 1 seconds, which is several orders of magnitude faster than conventional kernel search algorithms. We train our model using synthetic data generated from priors over a vocabulary of known kernels. By exploiting the nature of the self-attention mechanism, KITT is able to process datasets with inputs of arbitrary dimension. We demonstrate that kernels chosen by KITT yield strong performance over a diverse collection of regression benchmarks.

NeurIPS Conference 2021 Conference Paper

Marginalised Gaussian Processes with Nested Sampling

Fergus Simpson
Vidhi Lalchand
Carl Edward Rasmussen

Gaussian Process models are a rich distribution over functions with inductive biases controlled by a kernel function. Learning occurs through optimisation of the kernel hyperparameters using the marginal likelihood as the objective. This work proposes nested sampling as a means of marginalising kernel hyperparameters, because it is a technique that is well-suited to exploring complex, multi-modal distributions. We benchmark against Hamiltonian Monte Carlo on time-series and two-dimensional regression tasks, finding that a principled approach to quantifying hyperparameter uncertainty substantially improves the quality of prediction intervals.

JMLR Journal 2020 Journal Article

Convergence of Sparse Variational Inference in Gaussian Processes Regression

David R. Burt
Carl Edward Rasmussen
Mark van der Wilk

Gaussian processes are distributions over functions that are versatile and mathematically convenient priors in Bayesian modelling. However, their use is often impeded for data with large numbers of observations, $N$, due to the cubic (in $N$) cost of matrix operations used in exact inference. Many solutions have been proposed that rely on $M \ll N$ inducing variables to form an approximation at a cost of $\mathcal{O}\left(NM^2\right)$. While the computational cost appears linear in $N$, the true complexity depends on how $M$ must scale with $N$ to ensure a certain quality of the approximation. In this work, we investigate upper and lower bounds on how $M$ needs to grow with $N$ to ensure high quality approximations. We show that we can make the KL-divergence between the approximate model and the exact posterior arbitrarily small for a Gaussian-noise regression model with $M \ll N$. Specifically, for the popular squared exponential kernel and $D$-dimensional Gaussian distributed covariates, $M = \mathcal{O}((\log N)^D)$ suffice and a method with an overall computational cost of $\mathcal{O}\left(N(\log N)^{2D}(\log \log N)^2\right)$ can be used to perform inference. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2020. ( edit, beta )

NeurIPS Conference 2020 Conference Paper

Ensembling geophysical models with Bayesian Neural Networks

Ushnish Sengupta
Matt Amos
Scott Hosking
Carl Edward Rasmussen
Matthew Juniper
Paul Young

Ensembles of geophysical models improve projection accuracy and express uncertainties. We develop a novel data-driven ensembling strategy for combining geophysical models using Bayesian Neural Networks, which infers spatiotemporally varying model weights and bias while accounting for heteroscedastic uncertainties in the observations. This produces more accurate and uncertainty-aware projections without sacrificing interpretability. Applied to the prediction of total column ozone from an ensemble of 15 chemistry-climate models, we find that the Bayesian neural network ensemble (BayNNE) outperforms existing ensembling methods, achieving a 49. 4% reduction in RMSE for temporal extrapolation, and a 67. 4% reduction in RMSE for polar data voids, compared to a weighted mean. Uncertainty is also well-characterized, with 90. 6% of the data points in our extrapolation validation dataset lying within 2 standard deviations and 98. 5% within 3 standard deviations.

NeurIPS Conference 2017 Conference Paper

Convolutional Gaussian Processes

Mark van der Wilk
Carl Edward Rasmussen
James Hensman

We present a practical way of introducing convolutional structure into Gaussian processes, making them more suited to high-dimensional inputs like images. The main contribution of our work is the construction of an inter-domain inducing point approximation that is well-tailored to the convolutional kernel. This allows us to gain the generalisation benefit of a convolutional kernel, together with fast but accurate posterior inference. We investigate several variations of the convolutional kernel, and apply it to MNIST and CIFAR-10, where we obtain significant improvements over existing Gaussian process models. We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance. This illustration of the usefulness of the marginal likelihood may help automate discovering architectures in larger models.

NeurIPS Conference 2017 Conference Paper

Data-Efficient Reinforcement Learning in Continuous State-Action Gaussian-POMDPs

Rowan McAllister
Carl Edward Rasmussen

We present a data-efficient reinforcement learning method for continuous state-action systems under significant observation noise. Data-efficient solutions under small noise exist, such as PILCO which learns the cartpole swing-up task in 30s. PILCO evaluates policies by planning state-trajectories using a dynamics model. However, PILCO applies policies to the observed state, therefore planning in observation space. We extend PILCO with filtering to instead plan in belief space, consistent with partially observable Markov decisions process (POMDP) planning. This enables data-efficient learning under significant observation noise, outperforming more naive methods such as post-hoc application of a filter to policies optimised by the original (unfiltered) PILCO algorithm. We test our method on the cartpole swing-up task, which involves nonlinear dynamics and requires nonlinear control.

NeurIPS Conference 2016 Conference Paper

Understanding Probabilistic Sparse Gaussian Process Approximations

Matthias Bauer
Mark van der Wilk
Carl Edward Rasmussen

Good sparse approximations are essential for practical inference in Gaussian Processes as the computational cost of exact methods is prohibitive for large datasets. The Fully Independent Training Conditional (FITC) and the Variational Free Energy (VFE) approximations are two recent popular methods. Despite superficial similarities, these approximations have surprisingly different theoretical properties and behave differently in practice. We thoroughly investigate the two methods for regression both analytically and through illustrative examples, and draw conclusions to guide practical application.

NeurIPS Conference 2014 Conference Paper

Distributed Variational Inference in Sparse Gaussian Process Regression and Latent Variable Models

Yarin Gal
Mark van der Wilk
Carl Edward Rasmussen

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 the decoupling of the data given the inducing points to re-formulate the evidence lower bound in a Map-Reduce setting. We show that the inference scales well with data and computational resources, while preserving a balanced distribution of the load among the nodes. We further demonstrate the utility in scaling Gaussian processes to big data. We show that GP performance improves with increasing amounts of data in regression (on flight data with 2 million records) and latent variable modelling (on MNIST). The results show that GPs perform better than many common models often used for big data.

NeurIPS Conference 2014 Conference Paper

Variational Gaussian Process State-Space Models

Roger Frigola
Yutian Chen
Carl Edward Rasmussen

State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dynamical systems. In comparison to conventional parametric models, we offer the possibility to straightforwardly trade off model capacity and computational cost whilst avoiding overfitting. Our main algorithm uses a hybrid inference approach combining variational Bayes and sequential Monte Carlo. We also present stochastic variational inference and online learning approaches for fast learning with long time series.

NeurIPS Conference 2013 Conference Paper

Bayesian Inference and Learning in Gaussian Process State-Space Models with Particle MCMC

Roger Frigola
Fredrik Lindsten
Thomas Schön
Carl Edward Rasmussen

State-space models are successfully used in many areas of science, engineering and economics to model time series and dynamical systems. We present a fully Bayesian approach to inference and learning in nonlinear nonparametric state-space models. We place a Gaussian process prior over the transition dynamics, resulting in a flexible model able to capture complex dynamical phenomena. However, to enable efficient inference, we marginalize over the dynamics of the model and instead infer directly the joint smoothing distribution through the use of specially tailored Particle Markov Chain Monte Carlo samplers. Once an approximation of the smoothing distribution is computed, the state transition predictive distribution can be formulated analytically. We make use of sparse Gaussian process models to greatly reduce the computational complexity of the approach.

JMLR Journal 2010 Journal Article

Gaussian Processes for Machine Learning (GPML) Toolbox

Carl Edward Rasmussen
Hannes Nickisch

The GPML toolbox provides a wide range of functionality for Gaussian process (GP) inference and prediction. GPs are specified by mean and covariance functions; we offer a library of simple mean and covariance functions and mechanisms to compose more complex ones. Several likelihood functions are supported including Gaussian and heavy-tailed for regression as well as others suitable for classification. Finally, a range of inference methods is provided, including exact and variational inference, Expectation Propagation, and Laplace's method dealing with non-Gaussian likelihoods and FITC for dealing with large regression tasks. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2010. ( edit, beta )

JMLR Journal 2010 Journal Article

Sparse Spectrum Gaussian Process Regression

Miguel Lázaro-Gredilla
Joaquin Quiñnero-Candela
Carl Edward Rasmussen
Aníbal R. Figueiras-Vidal

We present a new sparse Gaussian Process (GP) model for regression. The key novel idea is to sparsify the spectral representation of the GP. This leads to a simple, practical algorithm for regression tasks. We compare the achievable trade-offs between predictive accuracy and computational requirements, and show that these are typically superior to existing state-of-the-art sparse approximations. We discuss both the weight space and function space representations, and note that the new construction implies priors over functions which are always stationary, and can approximate any covariance function in this class. [abs] [ pdf ][ bib ] &copy JMLR 2010. ( edit, beta )

JMLR Journal 2008 Journal Article

Approximations for Binary Gaussian Process Classification

Hannes Nickisch
Carl Edward Rasmussen

We provide a comprehensive overview of many recent algorithms for approximate inference in Gaussian process models for probabilistic binary classification. The relationships between several approaches are elucidated theoretically, and the properties of the different algorithms are corroborated by experimental results. We examine both 1) the quality of the predictive distributions and 2) the suitability of the different marginal likelihood approximations for model selection (selecting hyperparameters) and compare to a gold standard based on MCMC. Interestingly, some methods produce good predictive distributions although their marginal likelihood approximations are poor. Strong conclusions are drawn about the methods: The Expectation Propagation algorithm is almost always the method of choice unless the computational budget is very tight. We also extend existing methods in various ways, and provide unifying code implementing all approaches. [abs] [ pdf ][ bib ] &copy JMLR 2008. ( edit, beta )

JMLR Journal 2007 Journal Article

The Need for Open Source Software in Machine Learning

Sören Sonnenburg
Mikio L. Braun
Cheng Soon Ong
Samy Bengio
Leon Bottou
Geoffrey Holmes
Yann LeCun
Klaus-Robert Müller

Open source tools have recently reached a level of maturity which makes them suitable for building large-scale real-world systems. At the same time, the field of machine learning has developed a large body of powerful learning algorithms for diverse applications. However, the true potential of these methods is not used, since existing implementations are not openly shared, resulting in software with low usability, and weak interoperability. We argue that this situation can be significantly improved by increasing incentives for researchers to publish their software under an open source model. Additionally, we outline the problems authors are faced with when trying to publish algorithmic implementations of machine learning methods. We believe that a resource of peer reviewed software accompanied by short articles would be highly valuable to both the machine learning and the general scientific community. [abs] [ pdf ][ bib ] &copy JMLR 2007. ( edit, beta )

JMLR Journal 2005 Journal Article

A Unifying View of Sparse Approximate Gaussian Process Regression

Joaquin Quiñonero-Candela
Carl Edward Rasmussen

We provide a new unifying view, including all existing proper probabilistic sparse approximations for Gaussian process regression. Our approach relies on expressing the effective prior which the methods are using. This allows new insights to be gained, and highlights the relationship between existing methods. It also allows for a clear theoretically justified ranking of the closeness of the known approximations to the corresponding full GPs. Finally we point directly to designs of new better sparse approximations, combining the best of the existing strategies, within attractive computational constraints. [abs] [ pdf ][ bib ] &copy JMLR 2005. ( edit, beta )

JMLR Journal 2005 Journal Article

Assessing Approximate Inference for Binary Gaussian Process Classification

Malte Kuss
Carl Edward Rasmussen

Gaussian process priors can be used to define flexible, probabilistic classification models. Unfortunately exact Bayesian inference is analytically intractable and various approximation techniques have been proposed. In this work we review and compare Laplace's method and Expectation Propagation for approximate Bayesian inference in the binary Gaussian process classification model. We present a comprehensive comparison of the approximations, their predictive performance and marginal likelihood estimates to results obtained by MCMC sampling. We explain theoretically and corroborate empirically the advantages of Expectation Propagation compared to Laplace's method. [abs] [ pdf ][ bib ] &copy JMLR 2005. ( edit, beta )