On Coordinate Minimization of Convex Piecewise-Affine Functions
Tomas Werner

TL;DR
This paper generalizes message-passing algorithms for convex piecewise-affine functions, showing they converge to local consistency conditions, which are relaxations of global optimality, enhancing understanding of their convergence behavior.
Contribution
It introduces a generalized coordinate minimization method applicable to any convex piecewise-affine function, extending existing algorithms like max-sum diffusion.
Findings
Converges to a local consistency condition
Generalizes max-sum diffusion to arbitrary convex piecewise-affine functions
Provides a sign relaxation of the global optimality condition
Abstract
A popular class of algorithms to optimize the dual LP relaxation of the discrete energy minimization problem (a.k.a.\ MAP inference in graphical models or valued constraint satisfaction) are convergent message-passing algorithms, such as max-sum diffusion, TRW-S, MPLP and SRMP. These algorithms are successful in practice, despite the fact that they are a version of coordinate minimization applied to a convex piecewise-affine function, which is not guaranteed to converge to a global minimizer. These algorithms converge only to a local minimizer, characterized by local consistency known from constraint programming. We generalize max-sum diffusion to a version of coordinate minimization applicable to an arbitrary convex piecewise-affine function, which converges to a local consistency condition. This condition can be seen as the sign relaxation of the global optimality condition.
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Taxonomy
TopicsError Correcting Code Techniques · Constraint Satisfaction and Optimization · Machine Learning and Algorithms
