Advances in Scalable Bayesian Inference: Gaussian Processes & Discrete Variable Models

Gaussian processes (GPs) are exactly the models we would like to use for modern machine learning tasks; they are non-parametric models whose capacity naturally adapts to the quantity of training data, are highly interpretable, offer powerful opportunities to incorporate prior knowledge, and they deal with uncertainty due to lack of data in a rigorous manner through Bayesian inference. Unfortunately, the generic algorithm for GP training and inference scales with O(n^3) time and O(n^2) storage on a problem with n training observations. Given present-day computational resources, this scaling makes GPs struggle to scale beyond modestly sized datasets. This thesis explores approaches to scale Gaussian process training and inference to large datasets without sacrificing the benefits of these models. Specifically, we present theoretical analyses alongside algorithmic advances in GP modelling and inference for regression and classification problems. First, we consider GP inference on a problem structure present in many spatiotemporal problems and develop several algorithms that dramatically reduce the complexity of exact GP inference using Kronecker matrix algebra. We then discuss a novel GP approximation by showing how a highly accurate Nyström approximation of kernel eigenfunction can use as many as 10^33 inducing points with little computational expense. We subsequently consider a highly general class of GP covariance kernels and show that the GP marginal likelihood can be computed with a complexity that is independent of the number of training observations and as low as linear in the number of kernel basis functions. We then consider a variational inference approximation to the GP posterior that exploits a stochastic training strategy whose per iteration complexity is independent of both the number of training examples and the number of kernel basis functions. We show that this unique approach enables the use of high-capacity GP models on large datasets for regression and classification. Finally, we consider a discrete relaxation of continuous priors that enables fast inferencing on devices with limited computing resources. We develop a novel variational inference procedure that exploits Kronecker matrix algebra to compute the variational bound exactly and with a complexity that is independent of the dataset size.

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