Structured Prediction via the Extragradient Method

We present a simple and scalable algorithm for large-margin estimation of structured models, including an important class of Markov networks and combinatorial models. The estimation problem can be formulated as a quadratic program (QP) that exploits the problem structure to achieve polynomial number of variables and constraints. However, off-the-shelf QP solvers scale poorly with problem and training sample size. We recast the formulation as a convex-concave saddle point problem that allows us to use simple projection methods. We show the projection step can be solved using combinatorial algorithms for min-cost convex flow. We provide linear convergence guarantees for our method and present experiments on two very different structured prediction tasks: 3D image segmentation and word alignment, illustrating the favorable scaling properties of our algorithm.