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. This is also known as convex/convergent message passing; examples are max-sum diffusion and sequential tree-reweighted message passing (TRW-S). 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 point). We prove a stronger result (conjectured before but never proved): the iterates converge to a fixed point of the method. Moreover, we show that the algorithm terminates within $\mathcal{O}(1/\varepsilon)$ iterations. We first prove this for a version of coordinate descent applied to a general piecewise-affine convex objective. Then we show that several convex message passing methods are special cases of this method. Finally, we show that a slightly different version of coordinate descent can cycle.

翻译：在图模型的MAP推断问题中，一种流行方法是通过（块）坐标下降来最小化对偶线性规划或拉格朗日松弛得到的上界。这也被称为凸/收敛消息传递；例如最大和扩散与序列树重加权消息传递（TRW-S）。这些方法的收敛性质目前尚未被完全理解。已有证明表明它们会收敛到由活动约束局部一致性所刻画的集合，但收敛速率未知；然而，迭代序列是否收敛（到任意点）尚不明确。我们证明了一个更强的结果（此前曾被猜想但从未得到证明）：迭代序列会收敛到该方法的不动点。此外，我们证明算法在$\mathcal{O}(1/\varepsilon)$次迭代内终止。我们首先针对应用于一般分段仿射凸目标的坐标下降变体证明该结论，随后说明若干凸消息传递方法是该方法的特例。最后，我们证明坐标下降的一个略微不同的变体可能出现循环。