  Many types of data from fields including natural language processing,
computer vision, and bioinformatics, are well represented by discrete,
compositional structures such as trees, sequences, or matchings. Latent
structure models are a powerful tool for learning to extract such
representations, offering a way to incorporate structural bias, discover
insight about the data, and interpret decisions. However, effective training is
challenging, as neural networks are typically designed for continuous
computation.
  This text explores three broad strategies for learning with discrete latent
structure: continuous relaxation, surrogate gradients, and probabilistic
estimation. Our presentation relies on consistent notations for a wide range of
models. As such, we reveal many new connections between latent structure
learning strategies, showing how most consist of the same small set of
fundamental building blocks, but use them differently, leading to substantially
different applicability and properties.
