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.

Software packages like TensorFlow and PyTorch are designed to support linear algebra operations, and their speed and usability determine their success. However, by prioritising speed, they often neglect memory requirements. As a consequence, the implementations of memory-intensive algorithms that are convenient in terms of software design can often not be run for large problems due to memory overflows. Memory-efficient solutions require complex programming approaches with significant logic outside the computational framework. This impairs the adoption and use of such algorithms. To address this, we developed an XLA compiler extension that adjusts the computational data-flow representation of an algorithm according to a user-specified memory limit. We show that k-nearest neighbour, sparse Gaussian process regression methods and Transformers can be run on a single device at a much larger scale, where standard implementations would have failed. Our approach leads to better use of hardware resources. We believe that further focus on removing memory constraints at a compiler level will widen the range of machine learning methods that can be developed in the future.

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.

Thompson Sampling (TS) from Gaussian Process (GP) models is a powerful tool for the optimization of black-box functions. Although TS enjoys strong theoretical guarantees and convincing empirical performance, it incurs a large computational overhead that scales polynomially with the optimization budget. Recently, scalable TS methods based on sparse GP models have been proposed to increase the scope of TS, enabling its application to problems that are sufficiently multi-modal, noisy or combinatorial to require more than a few hundred evaluations to be solved. However, the approximation error introduced by sparse GPs invalidates all existing regret bounds. In this work, we perform a theoretical and empirical analysis of scalable TS. We provide theoretical guarantees and show that the drastic reduction in computational complexity of scalable TS can be enjoyed without loss in the regret performance over the standard TS. These conceptual claims are validated for practical implementations of scalable TS on synthetic benchmarks and as part of a real-world high-throughput molecular design task.

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.In this work, 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.

In decision-making systems, it is important to have classifiers that have calibrated uncertainties, with an optimisation objective that can be used for automated model selection and training. Gaussian processes (GPs) provide uncertainty estimates and a marginal likelihood objective, but their weak inductive biases lead to inferior accuracy. This has limited their applicability in certain tasks (e.g. image classification). We propose a translation insensitive convolutional kernel, which relaxes the translation invariance constraint imposed by previous convolutional GPs. We show how we can use the marginal likelihood to learn the degree of insensitivity. We also reformulate GP image-to-image convolutional mappings as multi-output GPs, leading to deep convolutional GPs. We show experimentally that our new kernel improves performance in both single-layer and deep models. We also demonstrate that our fully Bayesian approach improves on dropout-based Bayesian deep learning methods in terms of uncertainty and marginal likelihood estimates.