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Chris J. Oates

Publications and source records attributed to Chris J. Oates.

At least 19 recordsLinked to original sources
arXiv2026

Why not to use the Gaussian kernel

Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density.

stat.ML↗
arXiv2026

Extrapolation of Tempered Posteriors

Tempering is a popular tool in Bayesian computation, being used to transform a posterior distribution $p_1$ into a reference distribution $p_0$ that is more easily approximated. Several algorithms exist that start by approximating $p_0$ and proceed through a sequence of intermediate distributions $p_t$ until an approximation to $p_1$ is obtained. Our contribution reveals that high-quality approximation of terms up to $p_1$ is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish conditions under which posterior expectations are determined by their associated tempered expectations on any non-empty $t$ interval. Harnessing this result, we propose novel methodology for approximating posterior expectations based on extrapolation and smoothing of tempered expectations, which we implement as a post-processing variance-reduction tool for sequential Monte Carlo.

stat.CO↗
arXiv2026

Thinned Mean Field Langevin Dynamics

Several important learning tasks can be formulated as minimizing an entropy-regularized objective over an appropriate space of probability distributions. Mean-field Langevin dynamics (MFLD) facilitate computation in this general context, casting the minimizer as the invariant distribution of a McKean--Vlasov process, which can be numerically discretized using $N$ particles and thus simulated. However, simulating this interacting particle system has computational complexity of order $N^2$. Motivated by recent research into \emph{kernel thinning}, we propose \texttt{KT-MFLD}, in which each particle interacts only with a thinned particle coreset of size $\mathcal{O}(N^{\frac{1}{2}})$. \texttt{KT-MFLD} thus reduces the computational complexity to order $N^{\frac{3}{2}}$ while, under mild regularity conditions, achieving the same convergence guarantees (up to logarithmic factors) as MFLD. Our theoretical analysis is empirically confirmed on tasks including the training of student-teacher neural networks, quantization with maximum mean discrepancy, and computation of predictively-oriented posteriors in a post-Bayesian framework.

cs.LG↗
arXiv2025

Operator-Informed Score Matching for Markov Diffusion Models

Diffusion models are typically trained using score matching, a learning objective agnostic to the underlying noising process that guides the model. This paper argues that Markov noising processes enjoy an advantage over alternatives, as the Markov operators that govern the noising process are well-understood. Specifically, by leveraging the spectral decomposition of the infinitesimal generator of the Markov noising process, we obtain parametric estimates of the score functions simultaneously for all marginal distributions, using only sample averages with respect to the data distribution. The resulting operator-informed score matching provides both a standalone approach to sample generation for low-dimensional distributions, as well as a recipe for better informed neural score estimators in high-dimensional settings.

stat.ML↗
arXiv2024

Online Semiparametric Regression via Sequential Monte Carlo

We develop and describe online algorithms for performing online semiparametric regression analyses. Earlier work on this topic is in Luts, Broderick & Wand (J. Comput. Graph. Statist., 2014) where online mean field variational Bayes was employed. In this article we instead develop sequential Monte Carlo approaches to circumvent well-known inaccuracies inherent in variational approaches. Even though sequential Monte Carlo is not as fast as online mean field variational Bayes, it can be a viable alternative for applications where the data rate is not overly high. For Gaussian response semiparametric regression models our new algorithms share the online mean field variational Bayes property of only requiring updating and storage of sufficient statistics quantities of streaming data. In the non-Gaussian case accurate real-time semiparametric regression requires the full data to be kept in storage. The new algorithms allow for new options concerning accuracy/speed trade-offs for online semiparametric regression.

stat.ME↗
arXiv2024

Scalable Monte Carlo for Bayesian Learning

This book aims to provide a graduate-level introduction to advanced topics in Markov chain Monte Carlo (MCMC) algorithms, as applied broadly in the Bayesian computational context. Most, if not all of these topics (stochastic gradient MCMC, non-reversible MCMC, continuous time MCMC, and new techniques for convergence assessment) have emerged as recently as the last decade, and have driven substantial recent practical and theoretical advances in the field. A particular focus is on methods that are scalable with respect to either the amount of data, or the data dimension, motivated by the emerging high-priority application areas in machine learning and AI.

stat.ML↗
arXiv2023

Meta-learning Control Variates: Variance Reduction with Limited Data

Control variates can be a powerful tool to reduce the variance of Monte Carlo estimators, but constructing effective control variates can be challenging when the number of samples is small. In this paper, we show that when a large number of related integrals need to be computed, it is possible to leverage the similarity between these integration tasks to improve performance even when the number of samples per task is very small. Our approach, called meta learning CVs (Meta-CVs), can be used for up to hundreds or thousands of tasks. Our empirical assessment indicates that Meta-CVs can lead to significant variance reduction in such settings, and our theoretical analysis establishes general conditions under which Meta-CVs can be successfully trained.

stat.ME↗
arXiv2023

Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed

Gaussian process regression underpins countless academic and industrial applications of machine learning and statistics, with maximum likelihood estimation routinely used to select appropriate parameters for the covariance kernel. However, it remains an open problem to establish the circumstances in which maximum likelihood estimation is well-posed, that is, when the predictions of the regression model are insensitive to small perturbations of the data. This article identifies scenarios where the maximum likelihood estimator fails to be well-posed, in that the predictive distributions are not Lipschitz in the data with respect to the Hellinger distance. These failure cases occur in the noiseless data setting, for any Gaussian process with a stationary covariance function whose lengthscale parameter is estimated using maximum likelihood. Although the failure of maximum likelihood estimation is part of Gaussian process folklore, these rigorous theoretical results appear to be the first of their kind. The implication of these negative results is that well-posedness may need to be assessed post-hoc, on a case-by-case basis, when maximum likelihood estimation is used to train a Gaussian process model.

math.ST↗
arXiv2023

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

The Matérn model has been a cornerstone of spatial statistics for more than half a century. More recently, the Matérn model has been central to disciplines as diverse as numerical analysis, approximation theory, computational statistics, machine learning, and probability theory. In this article we take a Matérn-based journey across these disciplines. First, we reflect on the importance of the Matérn model for estimation and prediction in spatial statistics, establishing also connections to other disciplines in which the Matérn model has been influential. Then, we position the Matérn model within the literature on big data and scalable computation: the SPDE approach, the Vecchia likelihood approximation, and recent applications in Bayesian computation are all discussed. Finally, we review recent devlopments, including flexible alternatives to the Matérn model, whose performance we compare in terms of estimation, prediction, screening effect, computation, and Sobolev regularity properties.

math.ST↗
arXiv2022

BayesCG As An Uncertainty Aware Version of CG

The Bayesian Conjugate Gradient method (BayesCG) is a probabilistic generalization of the Conjugate Gradient method (CG) for solving linear systems with real symmetric positive definite coefficient matrices. Our CG-based implementation of BayesCG under a structure-exploiting prior distribution represents an 'uncertainty-aware' version of CG. Its output consists of CG iterates and posterior covariances that can be propagated to subsequent computations. The covariances have low-rank and are maintained in factored form. This allows easy generation of accurate samples to probe uncertainty in downstream computations. Numerical experiments confirm the effectiveness of the low-rank posterior covariances.

math.NA↗
arXiv2022

Regularized Zero-Variance Control Variates

Zero-variance control variates (ZV-CV) are a post-processing method to reduce the variance of Monte Carlo estimators of expectations using the derivatives of the log target. Once the derivatives are available, the only additional computational effort lies in solving a linear regression problem. Significant variance reductions have been achieved with this method in low dimensional examples, but the number of covariates in the regression rapidly increases with the dimension of the target. In this paper, we present compelling empirical evidence that the use of penalized regression techniques in the selection of high-dimensional control variates provides performance gains over the classical least squares method. Another type of regularization based on using subsets of derivatives, or a priori regularization as we refer to it in this paper, is also proposed to reduce computational and storage requirements. Several examples showing the utility and limitations of regularized ZV-CV for Bayesian inference are given. The methods proposed in this paper are accessible through the R package ZVCV.

stat.CO↗
arXiv2022

Statistical Properties of the Probabilistic Numeric Linear Solver BayesCG

We analyse the calibration of BayesCG under the Krylov prior, a probabilistic numeric extension of the Conjugate Gradient (CG) method for solving systems of linear equations with symmetric positive definite coefficient matrix. Calibration refers to the statistical quality of the posterior covariances produced by a solver. Since BayesCG is not calibrated in the strict existing notion, we propose instead two test statistics that are necessary but not sufficient for calibration: the Z-statistic and the new S-statistic. We show analytically and experimentally that under low-rank approximate Krylov posteriors, BayesCG exhibits desirable properties of a calibrated solver, is only slightly optimistic, and is computationally competitive with CG.

math.NA↗
arXiv2022

Testing whether a Learning Procedure is Calibrated

A learning procedure takes as input a dataset and performs inference for the parameters $θ$ of a model that is assumed to have given rise to the dataset. Here we consider learning procedures whose output is a probability distribution, representing uncertainty about $θ$ after seeing the dataset. Bayesian inference is a prime example of such a procedure, but one can also construct other learning procedures that return distributional output. This paper studies conditions for a learning procedure to be considered calibrated, in the sense that the true data-generating parameters are plausible as samples from its distributional output. A learning procedure whose inferences and predictions are systematically over- or under-confident will fail to be calibrated. On the other hand, a learning procedure that is calibrated need not be statistically efficient. A hypothesis-testing framework is developed in order to assess, using simulation, whether a learning procedure is calibrated. Several vignettes are presented to illustrate different aspects of the framework.

math.ST↗
arXiv2022

The Ridgelet Prior: A Covariance Function Approach to Prior Specification for Bayesian Neural Networks

Bayesian neural networks attempt to combine the strong predictive performance of neural networks with formal quantification of uncertainty associated with the predictive output in the Bayesian framework. However, it remains unclear how to endow the parameters of the network with a prior distribution that is meaningful when lifted into the output space of the network. A possible solution is proposed that enables the user to posit an appropriate Gaussian process covariance function for the task at hand. Our approach constructs a prior distribution for the parameters of the network, called a ridgelet prior, that approximates the posited Gaussian process in the output space of the network. In contrast to existing work on the connection between neural networks and Gaussian processes, our analysis is non-asymptotic, with finite sample-size error bounds provided. This establishes the universality property that a Bayesian neural network can approximate any Gaussian process whose covariance function is sufficiently regular. Our experimental assessment is limited to a proof-of-concept, where we demonstrate that the ridgelet prior can out-perform an unstructured prior on regression problems for which a suitable Gaussian process prior can be provided.

stat.ML↗
arXiv2021

Integration in reproducing kernel Hilbert spaces of Gaussian kernels

The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm for efficient numerical integration of functions reproduced by Gaussian kernels has not been fully solved. In this article we construct two classes of algorithms that use $N$ evaluations to integrate $d$-variate functions reproduced by Gaussian kernels and prove the exponential or super-algebraic decay of their worst-case errors. In contrast to earlier work, no constraints are placed on the length-scale parameter of the Gaussian kernel. The first class of algorithms is obtained via an appropriate scaling of the classical Gauss-Hermite rules. For these algorithms we derive lower and upper bounds on the worst-case error of the forms $\exp(-c_1 N^{1/d}) N^{1/(4d)}$ and $\exp(-c_2 N^{1/d}) N^{-1/(4d)}$, respectively, for positive constants $c_1 > c_2$. The second class of algorithms we construct is more flexible and uses worst-case optimal weights for points that may be taken as a nested sequence. For these algorithms we derive upper bounds of the form $\exp(-c_3 N^{1/(2d)})$ for a positive constant $c_3$.

math.NA↗
arXiv2021

Probabilistic Iterative Methods for Linear Systems

This paper presents a probabilistic perspective on iterative methods for approximating the solution $\mathbf{x}_* \in \mathbb{R}^d$ of a nonsingular linear system $\mathbf{A} \mathbf{x}_* = \mathbf{b}$. In the approach a standard iterative method on $\mathbb{R}^d$ is lifted to act on the space of probability distributions $\mathcal{P}(\mathbb{R}^d)$. Classically, an iterative method produces a sequence $\mathbf{x}_m$ of approximations that converge to $\mathbf{x}_*$. The output of the iterative methods proposed in this paper is, instead, a sequence of probability distributions $μ_m \in \mathcal{P}(\mathbb{R}^d)$. The distributional output both provides a "best guess" for $\mathbf{x}_*$, for example as the mean of $μ_m$, and also probabilistic uncertainty quantification for the value of $\mathbf{x}_*$ when it has not been exactly determined. Theoretical analysis is provided in the prototypical case of a stationary linear iterative method. In this setting we characterise both the rate of contraction of $μ_m$ to an atomic measure on $\mathbf{x}_*$ and the nature of the uncertainty quantification being provided. We conclude with an empirical illustration that highlights the insight into solution uncertainty that can be provided by probabilistic iterative methods.

stat.ME↗
arXiv2020

Measure Transport with Kernel Stein Discrepancy

Measure transport underpins several recent algorithms for posterior approximation in the Bayesian context, wherein a transport map is sought to minimise the Kullback--Leibler divergence (KLD) from the posterior to the approximation. The KLD is a strong mode of convergence, requiring absolute continuity of measures and placing restrictions on which transport maps can be permitted. Here we propose to minimise a kernel Stein discrepancy (KSD) instead, requiring only that the set of transport maps is dense in an $L^2$ sense and demonstrating how this condition can be validated. The consistency of the associated posterior approximation is established and empirical results suggest that KSD is competitive and more flexible alternative to KLD for measure transport.

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