1 Introduction to Dual Decomposition for Inference

Many inference problems with discrete variables result in a difficult com-binatorial optimization problem. In recent years, the technique of dual decomposition , also called Lagrangian relaxation, has proven to be a powerful means of solving these inference problems by decomposing them into simpler components that are repeatedly solved independently and combined into a global solution. In this chapter, we introduce the general technique of dual decomposition through its application to the problem of finding the most likely (MAP) assignment in Markov random fields. We discuss both subgradient and block coordinate descent approaches to solving the dual problem. The resulting message-passing algorithms are similar to max-product, but can be shown to solve a linear programming relaxation of the MAP problem. We show how many of the MAP algorithms are related to each other, and also quantify when the MAP solution can and cannot be decoded directly from the dual solution. 1.1 Introduction Many problems in engineering and the sciences require solutions to challenging combinatorial optimization problems. These include traditional problems such as scheduling, planning, fault diagnosis, or searching for molecular conformations. In addition, a wealth of combinatorial problems arise directly from probabilistic modeling (graphical models). Graphical models (see Koller and Friedman, 2009, for a textbook introduction) have been widely adopted in areas such as computational biology, machine vision, and natural language processing, and are increasingly being used as a framework for expressing combinatorial problems. Consider, for example, a protein side-chain placement problem where the goal is to find the minimum energy conformation of amino acid side-chains along a fixed carbon backbone. The orientations of the side-chains are represented by discretized angles called rotamers. The combinatorial difficulty arises here from the fact that rotamer choices for nearby amino acids are energetically coupled. For globular proteins, for example, such couplings may be present for most pairs of side-chain orientations. This problem is couched in probabilistic modeling terms by associating molecular conformations with the setting of discrete random variables corresponding to the rotamer angles. The interactions between such random variables come from the energetic couplings between nearby amino acids. Finding the minimum energy conformation is then equivalently solved by finding the most probable assignment of states to the variables. We will consider here combinatorial problems that are expressed in terms of structured probability models (graphical models). A graphical model is defined over a set of discrete variables x = {x j } j∈V. Local …