  As the size of datasets used in statistical learning continues to grow,
distributed training of models has attracted increasing attention. These
methods partition the data and exploit parallelism to reduce memory and
runtime, but suffer increasingly from communication costs as the data size or
the number of iterations grows. Recent work on linear models has shown that a
surrogate likelihood can be optimized locally to iteratively improve on an
initial solution in a communication-efficient manner. However, existing
versions of these methods experience multiple shortcomings as the data size
becomes massive, including diverging updates and efficiently handling sparsity.
In this work we develop solutions to these problems which enable us to learn a
communication-efficient distributed logistic regression model even beyond
millions of features. In our experiments we demonstrate a large improvement in
accuracy over distributed algorithms with only a few distributed update steps
needed, and similar or faster runtimes. Our code is available at
\url{https://github.com/FutureComputing4AI/ProxCSL}.
