Solving relaxations of MAP-MRF problems: Combinatorial in-face Frank-Wolfe directions

TL;DR

This paper introduces an efficient in-face Frank-Wolfe method for solving LP relaxations of MAP-MRF inference problems, significantly improving computational performance and establishing a new state-of-the-art in certain problem classes.

Contribution

It develops an in-face Frank-Wolfe algorithm tailored for MAP-MRF LP relaxations, enhancing efficiency over previous methods.

Findings

State-of-the-art performance on some MAP-MRF problems

Efficient implementation of in-face Frank-Wolfe directions

Open-source code available for reproducibility

Abstract

We consider the problem of solving LP relaxations of MAP-MRF inference problems, and in particular the method proposed recently in (Swoboda, Kolmogorov 2019; Kolmogorov, Pock 2021). As a key computational subroutine, it uses a variant of the Frank-Wolfe (FW) method to minimize a smooth convex function over a combinatorial polytope. We propose an efficient implementation of this subproutine based on in-face Frank-Wolfe directions, introduced in (Freund et al. 2017) in a different context. More generally, we define an abstract data structure for a combinatorial subproblem that enables in-face FW directions, and describe its specialization for tree-structured MAP-MRF inference subproblems. Experimental results indicate that the resulting method is the current state-of-art LP solver for some classes of problems. Our code is available at https://pub.ist.ac.at/~vnk/papers/IN-FACE-FW.html.