  In the deep learning (DL) era, parsing models are extremely simplified with
little hurt on performance, thanks to the remarkable capability of multi-layer
BiLSTMs in context representation. As the most popular graph-based dependency
parser due to its high efficiency and performance, the biaffine parser directly
scores single dependencies under the arc-factorization assumption, and adopts a
very simple local token-wise cross-entropy training loss. This paper for the
first time presents a second-order TreeCRF extension to the biaffine parser.
For a long time, the complexity and inefficiency of the inside-outside
algorithm hinder the popularity of TreeCRF. To address this issue, we propose
an effective way to batchify the inside and Viterbi algorithms for direct large
matrix operation on GPUs, and to avoid the complex outside algorithm via
efficient back-propagation. Experiments and analysis on 27 datasets from 13
languages clearly show that techniques developed before the DL era, such as
structural learning (global TreeCRF loss) and high-order modeling are still
useful, and can further boost parsing performance over the state-of-the-art
biaffine parser, especially for partially annotated training data. We release
our code at https://github.com/yzhangcs/crfpar.
