Many models in NLP involve latent variables, such as unknown parses, tags, or alignments. Finding the optimal model parameters is then usually a difficult nonconvex optimization problem. The usual practice is to settle for local optimization methods such as EM or gradient ascent. We explore how one might instead search for a global optimum in parameter space, using branch-and-bound. Our method would eventually find the global maximum (up to a user-specified ɛ) if run for long enough, but at any point can return a suboptimal solution together with an upper bound on the global maximum. As an illustrative case, we study a generative model for dependency parsing. We search for the maximum-likelihood model parameters and corpus parse, subject to posterior constraints. We show how to formulate this as a mixed integer quadratic programming problem with nonlinear constraints. We use the Reformulation Linearization Technique to produce convex relaxations during branch-and-bound. Although these techniques do not yet provide a practical solution to our instance of this NP-hard problem, they sometimes find better solutions than Viterbi EM with random restarts, in the same time.
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