A study of Nesterov's scheme for Lagrangian decomposition and MAP labeling

We study the MAP-labeling problem for graphical models by optimizing a dual problem obtained by Lagrangian decomposition. In this paper, we focus specifically on Nes-terov's optimal first-order optimization scheme for non-smooth convex programs, that has been studied for a range of other problems in computer vision and machine learning in recent years. We show that in order to obtain an efficiently convergent iteration, this approach should be augmented with a dynamic estimation of a corresponding Lip-schitz constant, leading to a runtime complexity of O(1/ϵ) in terms of the desired precision ϵ. Additionally, we devise a stopping criterion based on a duality gap as a sound basis for competitive comparison and show how to compute it efficiently. We evaluate our results using the publicly available Middlebury database and a set of computer generated graphical models that highlight specific aspects, along with other state-of-the-art methods for MAP-inference.