On Coordinate Minimization of Convex Piecewise-Affine Functions

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.