  We show how to train the fast dependency parser of Smith and Eisner (2008)
for improved accuracy. This parser can consider higher-order interactions among
edges while retaining O(n^3) runtime. It outputs the parse with maximum
expected recall -- but for speed, this expectation is taken under a posterior
distribution that is constructed only approximately, using loopy belief
propagation through structured factors. We show how to adjust the model
parameters to compensate for the errors introduced by this approximation, by
following the gradient of the actual loss on training data. We find this
gradient by back-propagation. That is, we treat the entire parser
(approximations and all) as a differentiable circuit, as Stoyanov et al. (2011)
and Domke (2010) did for loopy CRFs. The resulting trained parser obtains
higher accuracy with fewer iterations of belief propagation than one trained by
conditional log-likelihood.
