Convergence of Some Convex Message Passing Algorithms to a Fixed Point
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917684749271040 |
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| author | Voracek, Vaclav Werner, Tomas |
| author_facet | Voracek, Vaclav Werner, Tomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07004 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of Some Convex Message Passing Algorithms to a Fixed Point Voracek, Vaclav Werner, Tomas Artificial Intelligence Machine Learning Optimization and Control 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. |
| title | Convergence of Some Convex Message Passing Algorithms to a Fixed Point |
| topic | Artificial Intelligence Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2403.07004 |