Energy distribution view for monotonic dual decomposition

We consider the problem of finding the most probable explanation (also known as the MAP assignment) on probabilistic graphical models. The dual decomposition algorithms based on coordinate descent are efficient approximate techniques for this problem, where the local dual functions are constructed and optimized to monotonically increase the cost of the dual function. In this paper, we present a unifying framework for constructing and optimizing these local dual functions, and introduce an energy distribution view to analyze the convergence rates of these algorithms. To optimize the local dual functions, we first propose a new concept-the energy distribution ratio-to describe the features of the solutions, and then derive an explicit optimal solution, which covers most of the monotonic dual decomposition algorithms. It is shown that the differences of these algorithms lie in both the forms of the local dual functions and the settings of the energy distribution ratios, and the existing algorithms mainly focus on constructing compact and solvable local dual functions. In contrast, we study the impact of the energy distribution ratios and introduce two energy distribution criteria for fast convergence. Moreover, we exploit dynamic energy distribution ratios to optimize the local dual functions, and propose a series of improved algorithms. The experimental results on synthetic and real problems show the improved algorithms outperform the existing ones on the convergence performance.