  The automation of posterior inference in Bayesian data analysis has enabled
experts and nonexperts alike to use more sophisticated models, engage in faster
exploratory modeling and analysis, and ensure experimental reproducibility.
However, standard automated posterior inference algorithms are not tractable at
the scale of massive modern datasets, and modifications to make them so are
typically model-specific, require expert tuning, and can break theoretical
guarantees on inferential quality. Building on the Bayesian coresets framework,
this work instead takes advantage of data redundancy to shrink the dataset
itself as a preprocessing step, providing fully-automated, scalable Bayesian
inference with theoretical guarantees. We begin with an intuitive reformulation
of Bayesian coreset construction as sparse vector sum approximation, and
demonstrate that its automation and performance-based shortcomings arise from
the use of the supremum norm. To address these shortcomings we develop Hilbert
coresets, i.e., Bayesian coresets constructed under a norm induced by an
inner-product on the log-likelihood function space. We propose two Hilbert
coreset construction algorithms---one based on importance sampling, and one
based on the Frank-Wolfe algorithm---along with theoretical guarantees on
approximation quality as a function of coreset size. Since the exact
computation of the proposed inner-products is model-specific, we automate the
construction with a random finite-dimensional projection of the log-likelihood
functions. The resulting automated coreset construction algorithm is simple to
implement, and experiments on a variety of models with real and synthetic
datasets show that it provides high-quality posterior approximations and a
significant reduction in the computational cost of inference.
