Title: Learning Binary Trees via Sparse Relaxation. Abstract: One of the most classical problems in machine learning is how to learn binary trees that split data into meaningful partitions. From classification/regression via decision trees to hierarchical clustering, binary trees are useful because they (a) are often easy to visualize; (b) make computationally-efficient predictions; and (c) allow for flexible partitioning. Because of this there has been extensive research on how to learn such trees. Optimization generally falls into one of three categories: 1. greedy node-by-node optimization; 2. probabilistic relaxations for differentiability; 3. mixed-integer programming (MIP). Each of these have downsides: greedy can myopically choose poor splits, probabilistic relaxations do not have principled ways to prune trees, MIP methods can be slow on large problems and may not generalize. In this work we derive a novel sparse relaxation for binary tree learning. By sparsely relaxing a new MIP, our approach is able to learn tree splits and tree pruning using state-of-the-art gradient-based approaches. We demonstrate how our approach is easily visualizable, is efficient, and is competitive with current work in classification/regression and hierarchical clustering.