arXiv subjects

Nicolas Durrande

Publications and source records attributed to Nicolas Durrande.

At least 19 recordsLinked to original sources

arXiv2022

A Tutorial on Sparse Gaussian Processes and Variational Inference

Gaussian processes (GPs) provide a framework for Bayesian inference that can offer principled uncertainty estimates for a large range of problems. For example, if we consider regression problems with Gaussian likelihoods, a GP model enjoys a posterior in closed form. However, identifying the posterior GP scales cubically with the number of training examples and requires to store all examples in memory. In order to overcome these obstacles, sparse GPs have been proposed that approximate the true posterior GP with pseudo-training examples. Importantly, the number of pseudo-training examples is user-defined and enables control over computational and memory complexity. In the general case, sparse GPs do not enjoy closed-form solutions and one has to resort to approximate inference. In this context, a convenient choice for approximate inference is variational inference (VI), where the problem of Bayesian inference is cast as an optimization problem -- namely, to maximize a lower bound of the log marginal likelihood. This paves the way for a powerful and versatile framework, where pseudo-training examples are treated as optimization arguments of the approximate posterior that are jointly identified together with hyperparameters of the generative model (i.e. prior and likelihood). The framework can naturally handle a wide scope of supervised learning problems, ranging from regression with heteroscedastic and non-Gaussian likelihoods to classification problems with discrete labels, but also problems with multidimensional labels. The purpose of this tutorial is to provide access to the basic matter for readers without prior knowledge in both GPs and VI. A proper exposition to the subject enables also access to more recent advances (like importance-weighted VI as well as interdomain, multioutput and deep GPs) that can serve as an inspiration for new research ideas.

cs.LG↗

arXiv2022

Bayesian Quantile and Expectile Optimisation

Bayesian optimisation (BO) is widely used to optimise stochastic black box functions. While most BO approaches focus on optimising conditional expectations, many applications require risk-averse strategies and alternative criteria accounting for the distribution tails need to be considered. In this paper, we propose new variational models for Bayesian quantile and expectile regression that are well-suited for heteroscedastic noise settings. Our models consist of two latent Gaussian processes accounting respectively for the conditional quantile (or expectile) and the scale parameter of an asymmetric likelihood functions. Furthermore, we propose two BO strategies based on max-value entropy search and Thompson sampling, that are tailored to such models and that can accommodate large batches of points. Contrary to existing BO approaches for risk-averse optimisation, our strategies can directly optimise for the quantile and expectile, without requiring replicating observations or assuming a parametric form for the noise. As illustrated in the experimental section, the proposed approach clearly outperforms the state of the art in the heteroscedastic, non-Gaussian case.

stat.ML↗

arXiv2021

Deep Neural Networks as Point Estimates for Deep Gaussian Processes

Neural networks and Gaussian processes are complementary in their strengths and weaknesses. Having a better understanding of their relationship comes with the promise to make each method benefit from the strengths of the other. In this work, we establish an equivalence between the forward passes of neural networks and (deep) sparse Gaussian process models. The theory we develop is based on interpreting activation functions as interdomain inducing features through a rigorous analysis of the interplay between activation functions and kernels. This results in models that can either be seen as neural networks with improved uncertainty prediction or deep Gaussian processes with increased prediction accuracy. These claims are supported by experimental results on regression and classification datasets.

stat.ML↗

arXiv2021

Kernel Identification Through Transformers

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.

stat.ML↗

arXiv2021

The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain

In the univariate setting, using the kernel spectral representation is an appealing approach for generating stationary covariance functions. However, performing the same task for multiple-output Gaussian processes is substantially more challenging. We demonstrate that current approaches to modelling cross-covariances with a spectral mixture kernel possess a critical blind spot. For a given pair of processes, the cross-covariance is not reproducible across the full range of permitted correlations, aside from the special case where their spectral densities are of identical shape. We present a solution to this issue by replacing the conventional Gaussian components of a spectral mixture with block components of finite bandwidth (i.e. rectangular step functions). The proposed family of kernel represents the first multi-output generalisation of the spectral mixture kernel that can approximate any stationary multi-output kernel to arbitrary precision.

stat.ML↗

arXiv2020

Mat\'ern Gaussian Processes on Graphs

Gaussian processes are a versatile framework for learning unknown functions in a manner that permits one to utilize prior information about their properties. Although many different Gaussian process models are readily available when the input space is Euclidean, the choice is much more limited for Gaussian processes whose input space is an undirected graph. In this work, we leverage the stochastic partial differential equation characterization of Mat\'ern Gaussian processes - a widely-used model class in the Euclidean setting - to study their analog for undirected graphs. We show that the resulting Gaussian processes inherit various attractive properties of their Euclidean and Riemannian analogs and provide techniques that allow them to be trained using standard methods, such as inducing points. This enables graph Mat\'ern Gaussian processes to be employed in mini-batch and non-conjugate settings, thereby making them more accessible to practitioners and easier to deploy within larger learning frameworks.

stat.ML↗

arXiv2020

Sparse Gaussian Processes with Spherical Harmonic Features

We introduce a new class of inter-domain variational Gaussian processes (GP) where data is mapped onto the unit hypersphere in order to use spherical harmonic representations. Our inference scheme is comparable to variational Fourier features, but it does not suffer from the curse of dimensionality, and leads to diagonal covariance matrices between inducing variables. This enables a speed-up in inference, because it bypasses the need to invert large covariance matrices. Our experiments show that our model is able to fit a regression model for a dataset with 6 million entries two orders of magnitude faster compared to standard sparse GPs, while retaining state of the art accuracy. We also demonstrate competitive performance on classification with non-conjugate likelihoods.

stat.ML↗

arXiv2020

Automatic Tuning of Stochastic Gradient Descent with Bayesian Optimisation

Many machine learning models require a training procedure based on running stochastic gradient descent. A key element for the efficiency of those algorithms is the choice of the learning rate schedule. While finding good learning rates schedules using Bayesian optimisation has been tackled by several authors, adapting it dynamically in a data-driven way is an open question. This is of high practical importance to users that need to train a single, expensive model. To tackle this problem, we introduce an original probabilistic model for traces of optimisers, based on latent Gaussian processes and an auto-/regressive formulation, that flexibly adjusts to abrupt changes of behaviours induced by new learning rate values. As illustrated, this model is well-suited to tackle a set of problems: first, for the on-line adaptation of the learning rate for a cold-started run; then, for tuning the schedule for a set of similar tasks (in a classical BO setup), as well as warm-starting it for a new task.

stat.ML↗

arXiv2020

Doubly Sparse Variational Gaussian Processes

The use of Gaussian process models is typically limited to datasets with a few tens of thousands of observations due to their complexity and memory footprint. The two most commonly used methods to overcome this limitation are 1) the variational sparse approximation which relies on inducing points and 2) the state-space equivalent formulation of Gaussian processes which can be seen as exploiting some sparsity in the precision matrix. We propose to take the best of both worlds: we show that the inducing point framework is still valid for state space models and that it can bring further computational and memory savings. Furthermore, we provide the natural gradient formulation for the proposed variational parameterisation. Finally, this work makes it possible to use the state-space formulation inside deep Gaussian process models as illustrated in one of the experiments.

stat.ML↗

arXiv2019

Approximating Gaussian Process Emulators with Linear Inequality Constraints and Noisy Observations via MC and MCMC

Adding inequality constraints (e.g. boundedness, monotonicity, convexity) into Gaussian processes (GPs) can lead to more realistic stochastic emulators. Due to the truncated Gaussianity of the posterior, its distribution has to be approximated. In this work, we consider Monte Carlo (MC) and Markov Chain Monte Carlo (MCMC) methods. However, strictly interpolating the observations may entail expensive computations due to highly restrictive sample spaces. Furthermore, having (constrained) GP emulators when data are actually noisy is also of interest for real-world implementations. Hence, we introduce a noise term for the relaxation of the interpolation conditions, and we develop the corresponding approximation of GP emulators under linear inequality constraints. We show with various toy examples that the performance of MC and MCMC samplers improves when considering noisy observations. Finally, on 2D and 5D coastal flooding applications, we show that more flexible and realistic GP implementations can be obtained by considering noise effects and by enforcing the (linear) inequality constraints.

stat.ML↗

arXiv2019

Physically-Inspired Gaussian Process Models for Post-Transcriptional Regulation in Drosophila

The regulatory process of Drosophila is thoroughly studied for understanding a great variety of biological principles. While pattern-forming gene networks are analysed in the transcription step, post-transcriptional events (e.g. translation, protein processing) play an important role in establishing protein expression patterns and levels. Since the post-transcriptional regulation of Drosophila depends on spatiotemporal interactions between mRNAs and gap proteins, proper physically-inspired stochastic models are required to study the link between both quantities. Previous research attempts have shown that using Gaussian processes (GPs) and differential equations lead to promising predictions when analysing regulatory networks. Here we aim at further investigating two types of physically-inspired GP models based on a reaction-diffusion equation where the main difference lies in where the prior is placed. While one of them has been studied previously using protein data only, the other is novel and yields a simple approach requiring only the differentiation of kernel functions. In contrast to other stochastic frameworks, discretising the spatial space is not required here. Both GP models are tested under different conditions depending on the availability of gap gene mRNA expression data. Finally, their performances are assessed on a high-resolution dataset describing the blastoderm stage of the early embryo of Drosophila melanogaster

stat.ML↗

arXiv2019

Gaussian Process Modulated Cox Processes under Linear Inequality Constraints

Gaussian process (GP) modulated Cox processes are widely used to model point patterns. Existing approaches require a mapping (link function) between the unconstrained GP and the positive intensity function. This commonly yields solutions that do not have a closed form or that are restricted to specific covariance functions. We introduce a novel finite approximation of GP-modulated Cox processes where positiveness conditions can be imposed directly on the GP, with no restrictions on the covariance function. Our approach can also ensure other types of inequality constraints (e.g. monotonicity, convexity), resulting in more versatile models that can be used for other classes of point processes (e.g. renewal processes). We demonstrate on both synthetic and real-world data that our framework accurately infers the intensity functions. Where monotonicity is a feature of the process, our ability to include this in the inference improves results.

stat.ML↗

arXiv2019

Banded Matrix Operators for Gaussian Markov Models in the Automatic Differentiation Era

Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with banded precision. We show that this can efficiently be achieved by equipping an automatic differentiation framework, such as TensorFlow or PyTorch, with some linear algebra operators dedicated to banded matrices. This paper studies the algorithmic aspects of the required operators, details their reverse-mode derivatives, and show that their complexity is linear in the number of observations.

stat.ML↗

arXiv2018

Scalable GAM using sparse variational Gaussian processes

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.

cs.LG↗

arXiv2018

Bayesian inversion of a diffusion evolution equation with application to Biology

A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are considered, such as partial differential equations, this task may become challenging and ill-posed. In this work, a linear parabolic equation is considered where the objective is to estimate both the differential operator coefficients and the source term at once. The Bayesian methodology for inverse problems provides a form of regularization while quantifying uncertainty as the solution is a probability measure taking in account data. This posterior distribution, which is non-Gaussian and infinite dimensional, is then summarized through a mode and sampled using a state-of-the-art Markov-Chain Monte-Carlo algorithm based on a geometric approach. After a rigorous analysis, this methodology is applied on a dataset of the post-transcriptional regulation of Kni gap gene in the early development of Drosophila Melanogaster where mRNA concentration and both diffusion and depletion rates are inferred from noisy measurement of the protein concentration

math.NA↗

arXiv2017

Variational Fourier features for Gaussian processes

This work brings together two powerful concepts in Gaussian processes: the variational approach to sparse approximation and the spectral representation of Gaussian processes. This gives rise to an approximation that inherits the benefits of the variational approach but with the representational power and computational scalability of spectral representations. The work hinges on a key result that there exist spectral features related to a finite domain of the Gaussian process which exhibit almost-independent covariances. We derive these expressions for Matern kernels in one dimension, and generalize to more dimensions using kernels with specific structures. Under the assumption of additive Gaussian noise, our method requires only a single pass through the dataset, making for very fast and accurate computation. We fit a model to 4 million training points in just a few minutes on a standard laptop. With non-conjugate likelihoods, our MCMC scheme reduces the cost of computation from O(NM2) (for a sparse Gaussian process) to O(NM) per iteration, where N is the number of data and M is the number of features.

stat.ML↗

arXiv2017

Finite-dimensional Gaussian approximation with linear inequality constraints

Introducing inequality constraints in Gaussian process (GP) models can lead to more realistic uncertainties in learning a great variety of real-world problems. We consider the finite-dimensional Gaussian approach from Maatouk and Bay (2017) which can satisfy inequality conditions everywhere (either boundedness, monotonicity or convexity). Our contributions are threefold. First, we extend their approach in order to deal with general sets of linear inequalities. Second, we explore several Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution. Third, we investigate theoretical and numerical properties of the constrained likelihood for covariance parameter estimation. According to experiments on both artificial and real data, our full framework together with a Hamiltonian Monte Carlo-based sampler provides efficient results on both data fitting and uncertainty quantification.

stat.ML↗

arXiv2017

Properties and comparison of some Kriging sub-model aggregation methods

Kriging is a widely employed technique, in particular for computer experiments, in machine learning or in geostatistics. An important challenge for Kriging is the computational burden when the data set is large. This article focuses on a class of methods aiming at decreasing this computational cost, consisting in aggregating Kriging predictors based on smaller data subsets. It proves that aggregation methods that ignore the covariancebetween sub-models can yield an inconsistent final Kriging prediction. In contrast, a theoretical study of the nested Kriging method shows additional attractive properties for it: First, this predictor is consistent, second it can be interpreted as an exact conditional distribution for a modified process and third, the conditional covariances given the observations can be computed efficiently. This article also includes a theoretical and numerical analysis of how the assignment of the observation points to the sub-models can affect the prediction ability of the aggregated model. Finally, the nested Kriging method is extended to measurement errors and to universal Kriging.

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