Automated inference algorithms in Bayesian statistics have provided practitioners newfound access to fast, reproducible data analysis. But designing automated methods that are also computationally scalable and theoretically sound remains a significant challenge. Bayesian coresets takes the approach of compressing the dataset before running inference, providing scalability and guarantees on posterior approximation error. But the automation of past coreset methods is limited; they depend on the availability of a coarse posterior approximation, which is difficult to specify. In the present work we remove this requirement by formulating coreset construction as sparsity-constrained variational inference. This perspective leads to a novel construction via greedy optimization, and also provides a unifying information-geometric view of coreset methods. The proposed coreset construction algorithm is fully automated, requiring no problem-specific inputs aside from the probabilistic model and dataset. In addition to being significantly easier to use than past methods, experiments demonstrate that the proposed algorithm provides state-of-the-art Bayesian coreset constructions.