Convergence of Some Convex Message Passing Algorithms to a Fixed Point

Vaclav Voracek; Tomas Werner

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arXiv · 2403.07004

Convergence of Some Convex Message Passing Algorithms to a Fixed Point

Vaclav Voracek·Tomas Werner

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Abstract

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.

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Vaclav Voracek, Tomas Werner. 2024-03-07. Convergence of Some Convex Message Passing Algorithms to a Fixed Point. https://arxiv.org/abs/2403.07004

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