Mark van der Wilk
now a Associate Professor at University of Oxford
Publications
Tighter Bounds on the Log Marginal Likelihood of Gaussian Process Regression Using Conjugate Gradients
Artem Artemev, David R. Burt, Mark van der Wilk, 2021. (In 38th International Conference on Machine Learning).
Abstract▼ URL
We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model parameters by maximising our lower bound retains many benefits of the sparse variational approach while reducing the bias introduced into hyperparameter learning. The basis of our bound is a more careful analysis of the log-determinant term appearing in the log marginal likelihood, as well as using the method of conjugate gradients to derive tight lower bounds on the term involving a quadratic form. Our approach is a step forward in unifying methods relying on lower bound maximisation (e.g. variational methods) and iterative approaches based on conjugate gradients for training Gaussian processes. In experiments, we show improved predictive performance with our model for a comparable amount of training time compared to other conjugate gradient based approaches.
Understanding Probabilistic Sparse Gaussian Process Approximations
Matthias Stephan Bauer, Mark van der Wilk, Carl Edward Rasmussen, 2016. (In Advances in Neural Information Processing Systems 29).
Abstract▼ URL
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.
Comment: arXiv
Understanding variational inference in function-space
David R. Burt, Sebastian W. Ober, Adrià Garriga-Alonso, Mark van der Wilk, 2021. (In 3rd Symposium on Advances in Approximate Bayesian Inference).
Abstract▼ URL
Recent work has attempted to directly approximate the ‘function-space’ or predictive posterior distribution of Bayesian models, without approximating the posterior distribution over the parameters. This is appealing in e.g. Bayesian neural networks, where we only need the former, and the latter is hard to represent. In this work, we highlight some advantages and limitations of employing the Kullback-Leibler divergence in this setting. For example, we show that minimizing the KL divergence between a wide class of parametric distributions and the posterior induced by a (non-degenerate) Gaussian process prior leads to an ill-defined objective function. Then, we propose (featurized) Bayesian linear regression as a benchmark for ‘function-space’ inference methods that directly measures approximation quality. We apply this methodology to assess aspects of the objective function and inference scheme considered in Sun et al. (2018), emphasizing the quality of approximation to Bayesian inference as opposed to predictive performance.
Rates of Convergence for Sparse Variational Gaussian Process Regression
David R Burt, Carl Edward Rasmussen, Mark van der Wilk, 2019. (arXiv).
Abstract▼ URL
Excellent variational approximations to Gaussian process posteriors have been developed which avoid the O(N3) scaling with dataset size N. They reduce the computational cost to O(NM2), with M ≪N being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in N, the true complexity of the algorithm depends on how M must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We show that with high probability the KL divergence can be made arbitrarily small by growing M more slowly than N. A particular case of interest is that for regression with normally distributed inputs in D-dimensions with the popular Squared Exponential kernel, M=O(logDN) is sufficient. Our results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase M in continual learning scenarios.
Convergence of Sparse Variational Inference in Gaussian Processes Regression
David R. Burt, Carl Edward Rasmussen, Mark van der Wilk, 2020. (Journal of Machine Learning Research).
Abstract▼ URL
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 2). 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 D) suffice and a method with an overall computational cost of O(N(log N)2D(log log N)2) can be used to perform inference.
Deep Neural Networks as Point Estimates for Deep Gaussian Processes
Vincent Dutordoir, James Hensman, Mark van der Wilk, Carl Henrik Ek, Zoubin Ghahramani, Nicolas Durrande, Dec 2021. (In Advances in Neural Information Processing Systems 34). Online.
Abstract▼ URL
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.
Bayesian neural network priors revisited
Vincent Fortuin, Adrià Garriga-Alonso, Sebastian W. Ober, Florian Wenzel, Gunnar Rätsch, Richard E. Turner, Mark van der Wilk, Laurence Aitchison, 2022. (In 10th International Conference on Learning Representations).
Abstract▼ URL
Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. To find better priors, we study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observations into priors can lead to improved performance on a variety of image classification datasets. Surprisingly, these priors mitigate the cold posterior effect in FCNNs, but slightly increase the cold posterior effect in ResNets.
Distributed Variational Inference in Sparse Gaussian Process Regression and Latent Variable Models
Yarin Gal, Mark van der Wilk, Carl Rasmussen, 2014. (In Advances in Neural Information Processing Systems 27). Edited by Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence, K.Q. Weinberger. Curran Associates, Inc..
Abstract▼ URL
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.
Non-Factorised Variational Inference in Dynamical Systems
Alessandro Davide Ialongo, Mark van der Wilk, James Hensman, Carl Edward Rasmussen, December 2018. (In First Symposium on Advances in Approximate Bayesian Inference). Montreal.
Abstract▼ URL
We focus on variational inference in dynamical systems where the discrete time transition function (or evolution rule) is modelled by a Gaussian process. The dominant approach so far has been to use a factorised posterior distribution, decoupling the transition function from the system states. This is not exact in general and can lead to an overconfident posterior over the transition function as well as an overestimation of the intrinsic stochasticity of the system (process noise). We propose a new method that addresses these issues and incurs no additional computational costs.
Overcoming Mean-Field Approximations in Recurrent Gaussian Process Models
Alessandro Davide Ialongo, Mark van der Wilk, James Hensman, Carl Edward Rasmussen, June 2019. (In 36th International Conference on Machine Learning). Long Beach.
Abstract▼ URL
We identify a new variational inference scheme for dynamical systems whose transition function is modelled by a Gaussian process. Inference in this setting has either employed computationally intensive MCMC methods, or relied on factorisations of the variational posterior. As we demonstrate in our experiments, the factorisation between latent system states and transition function can lead to a miscalibrated posterior and to learning unnecessarily large noise terms. We eliminate this factorisation by explicitly modelling the dependence between state trajectories and the Gaussian process posterior. Samples of the latent states can then be tractably generated by conditioning on this representation. The method we obtain (VCDT: variationally coupled dynamics and trajectories) gives better predictive performance and more calibrated estimates of the transition function, yet maintains the same time and space complexities as mean-field methods. Code is available at: https://github.com/ialong/GPt.
Closed-form Inference and Prediction in Gaussian Process State-Space Models
Alessandro Davide Ialongo, Mark van der Wilk, Carl Edward Rasmussen, December 2017. (In NIPS Time Series Workshop 2017). Long Beach.
Abstract▼ URL
We examine an analytic variational inference scheme for the Gaussian Process State Space Model (GPSSM) - a probabilistic model for system identification and time-series modelling. Our approach performs variational inference over both the system states and the transition function. We exploit Markov structure in the true posterior, as well as an inducing point approximation to achieve linear time complexity in the length of the time series. Contrary to previous approaches, no Monte Carlo sampling is required: inference is cast as a deterministic optimisation problem. In a number of experiments, we demonstrate the ability to model non-linear dynamics in the presence of both process and observation noise as well as to impute missing information (e.g. velocities from raw positions through time), to de-noise, and to estimate the underlying dimensionality of the system. Finally, we also introduce a closed-form method for multi-step prediction, and a novel criterion for assessing the quality of our approximate posterior.
Concrete Problems for Autonomous Vehicle Safety: Advantages of Bayesian Deep Learning,
Rowan McAllister, Yarin Gal, Alex Kendall, Mark van der Wilk, Amar Shah, Roberto Cipolla, Adrian Weller, August 2017. (In International Joint Conference on Artificial Intelligence). Melbourne, Australia.
Abstract▼ URL
Autonomous vehicle (AV) software is typically composed of a pipeline of individual components, linking sensor inputs to motor outputs. Erroneous component outputs propagate downstream, hence safe AV software must consider the ultimate effect of each component’s errors. Further, improving safety alone is not sufficient. Passengers must also feel safe to trust and use AV systems. To address such concerns, we investigate three under-explored themes for AV research: safety, interpretability, and compliance. Safety can be improved by quantifying the uncertainties of component outputs and propagating them forward through the pipeline. Interpretability is concerned with explaining what the AV observes and why it makes the decisions it does, building reassurance with the passenger. Compliance refers to maintaining some control for the passenger. We discuss open challenges for research within these themes. We highlight the need for concrete evaluation metrics, propose example problems, and highlight possible solutions.
The promises and pitfalls of deep kernel learning
Sebastian W. Ober, Carl Edward Rasmussen, Mark van der Wilk, 2021. (In 37th Conference on Uncertainty in Artificial Intelligence).
Abstract▼ URL
Deep kernel learning (DKL) and related techniques aim to combine the representational power of neural networks with the reliable uncertainty estimates of Gaussian processes. One crucial aspect of these models is an expectation that, because they are treated as Gaussian process models optimized using the marginal likelihood, they are protected from overfitting. However, we identify situations where this is not the case. We explore this behavior, explain its origins and consider how it applies to real datasets. Through careful experimentation on the UCI, CIFAR-10, and the UTKFace datasets, we find that the overfitting from overparameterized maximum marginal likelihood, in which the model is “somewhat Bayesian”, can in certain scenarios be worse than that from not being Bayesian at all. We explain how and when DKL can still be successful by investigating optimization dynamics. We also find that failures of DKL can be rectified by a fully Bayesian treatment, which leads to the desired performance improvements over standard neural networks and Gaussian processes.
Last layer marginal likelihood for invariance learning
Pola E. Schwöbel, Martin Jørgensen, Sebastian W. Ober, Mark van der Wilk, 2022. (In 25th International Conference on Artificial Intelligence and Statistics).
Abstract▼ URL
Data augmentation is often used to incorporate inductive biases into models. Traditionally, these are hand-crafted and tuned with cross validation. The Bayesian paradigm for model selection provides a path towards end-to-end learning of invariances using only the training data, by optimising the marginal likelihood. Computing the marginal likelihood is hard for neural networks, but success with tractable approaches that compute the marginal likelihood for the last layer only raises the question of whether this convenient approach might be employed for learning invariances. We show partial success on standard benchmarks, in the low-data regime and on a medical imaging dataset by designing a custom optimisation routine. Introducing a new lower bound to the marginal likelihood allows us to perform inference for a larger class of likelihood functions than before. On the other hand, we demonstrate failure modes on the CIFAR10 dataset, where the last layer approximation is not sufficient due to the increased complexity of our neural network. Our results indicate that once more sophisticated approximations become available the marginal likelihood is a promising approach for invariance learning in neural networks.
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, 2022. (arXiv).
Abstract▼ URL
As Gaussian processes mature, they are increasingly being 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 the system. In this work, we study the numerical stability of scalable sparse approximations based on inducing points. 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 evaluate the proposed techniques on a number of examples, showing that, in geospatial settings, sparse approximations with guaranteed numerical stability often perform comparably to those without.
Convolutional Gaussian Processes
Mark van der Wilk, Carl Edward Rasmussen, James Hensman, 2017. (In Advances in Neural Information Processing Systems 31).
Abstract▼ URL
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, which have both been known to be challenging for Gaussian processes. We also show how the marginal likelihood can be used to find an optimal weighting between convolutional and RBF kernels to further improve performance. We hope that this illustration of the usefulness of a marginal likelihood will help automate discovering architectures in larger models.
Comment: arXiv