In modern data analysis, it is often useful to reduce the complexity of a large dataset by clustering the observations into a small and interpretable collection of subpopulations. Broadly speaking, there are two major approaches. In "model-based" clustering, the data are assumed to be generated by a (usually small) collection of simple probability distributions such as normal distributions, and clusters are inferred by fitting a probabilistic mixture model. Because of their transparent probabilistic assumptions, the statistical properties of mixture models are well-understood. In particular, if there is no model misspecification, i.e., the data truly come from a mixture distribution, then the subpopulations can be consistently estimated. Unfortunately, this appealing asymptotic guarantee is somewhat at odds with what is often observed in practice, whereby mixture models fitted to complex datasets often return an uninterpretably large number of components, many of which are quite similar to each other. The tendency of mixture models to overfit on real data leads many analysts to employ "model-free" clustering methods instead. A well-known example is hierarchical clustering, which organizes the data into a nested sequence of partitions at different resolutions. It is particularly useful for data exploration as it does not require fixing a number of subpopulations a priori and can be visualized using a dendrogram.

A common workflow in data exploration is to learn a low-dimensional representation of the data, identify groups of points in that representation, and examine the differences between the groups to determine what they represent. We treat this as an interpretable machine learning problem by leveraging the model that learned the low-dimensional representation to help identify the key differences between the groups. To solve this problem, we introduce a new type of explanation, a Global Counterfactual Explanation (GCE), and our algorithm, Transitive Global Translations (TGT), for computing GCEs. TGT identifies the differences between each pair of groups using compressed sensing but constrains those pairwise differences to be consistent among all of the groups. Empirically, we demonstrate that TGT is able to identify explanations that accurately explain the model while being relatively sparse, and that these explanations match real patterns in the data.

Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, COCOA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA algorithms, COCOA converges to the same .001-accurate solution quality on average 25× as quickly.

Communication remains the most significant bottleneck in the performance of distributed optimization algorithms for large-scale machine learning. In this paper, we propose a communication-efficient framework, CoCoA, that uses local computation in a primal-dual setting to dramatically reduce the amount of necessary communication. We provide a strong convergence rate analysis for this class of algorithms, as well as experiments on real-world distributed datasets with implementations in Spark. In our experiments, we find that as compared to state-of-the-art mini-batch versions of SGD and SDCA algorithms, CoCoA converges to the same .001-accurate solution quality on average 25x as quickly.