A popular approach to the MAP inference problem in graphical models is to minimize an upper bound obtained from a dual linear programming or Lagrangian relaxation by (block-)coordinate descent. Examples of such algorithms are max-sum diffusion and sequential tree-reweighted message passing. Convergence properties of these methods are currently not fully understood. They have been proved to converge to the set characterized by local consistency of active constraints, with unknown convergence rate; however, it was not clear if the iterates converge at all (to any single point). We prove a stronger result (which was conjectured before but never proved): the iterates converge to a fixed point of the algorithm. Moreover, we show that they achieve precision $\varepsilon>0$ in $\mathcal{O}(1/\varepsilon)$ iterations. We first prove this for a version of coordinate descent applied to a general piecewise-affine convex objective, using a novel proof technique. Then we demonstrate the generality of this approach by reducing some popular coordinate-descent algorithms to this problem. Finally we show that, in contrast to our main result, a similar version of coordinate descent applied to a constrained optimization problem need not converge.

It is known that fixed points of loopy belief propagation (BP) correspond to stationary points of the Bethe variational problem, where we minimize the Bethe free energy subject to normalization and marginalization constraints. Unfortunately, this does not entirely explain BP because BP is a dual rather than primal algorithm to solve the Bethe variational problem -- beliefs are infeasible before convergence. Thus, we have no better understanding of BP than as an algorithm to seek for a common zero of a system of non-linear functions, not explicitly related to each other. In this theoretical paper, we show that these functions are in fact explicitly related -- they are the partial derivatives of a single function of reparameterizations. That means, BP seeks for a stationary point of a single function, without any constraints. This function has a very natural form: it is a linear combination of local log-partition functions, exactly as the Bethe entropy is the same linear combination of local entropies.