Minimum Error Rate Training and the Convex Hull Semiring

We describe the line search used in the minimum error rate training algorithm MERT as the "inside score" of a weighted proof forest under a semiring defined in terms of well-understood operations from computational geometry. This conception leads to a straightforward complexity analysis of the dynamic programming MERT algorithms of Macherey et al. (2008) and Kumar et al. (2009) and practical approaches to implementation.