Multi-task learning (MTL) allows deep neural networks to learn from related tasks by sharing parameters with other networks. In practice, however, MTL involves searching an enormous space of possible parameter sharing architectures to find (a) the layers or subspaces that benefit from sharing, (b) the appropriate amount of sharing, and (c) the appropriate relative weights of the different task losses. Recent work has addressed each of the above problems in isolation. In this work we present an approach that learns a latent multi-task architecture that jointly addresses (a)--(c). We present experiments on synthetic data and data from OntoNotes 5.0, including four different tasks and seven different domains. Our extension consistently outperforms previous approaches to learning latent architectures for multi-task problems and achieves up to 15% average error reductions over common approaches to MTL.