Evaluation of a First-Order Primal-Dual Algorithm for MRF Energy Minimization

We investigate the First-Order Primal-Dual (FPD) algorithm of Chambolle and Pock [1] in connection with MAP inference for general discrete graphical models. We provide a tight analytical upper bound of the stepsize parameter as a function of the underlying graphical structure (number of states, graph connectivity) and thus insight into the dependency of the convergence rate on the problem structure. Furthermore, we provide a method to compute efficiently primal and dual feasible solutions as part of the FPD iteration, which allows to obtain a sound termination criterion based on the primal-dual gap. An experimental comparison with Nesterov’s first-order method in connection with dual decomposition shows superiority of the latter one in optimizing the dual problem. However due to the direct optimization of the primal bound, for small-sized (e.g. 20x20 grid graphs) problems with a large number of states, FPD iterations lead to faster improvement of the primal bound and a resulting faster overall convergence.