Parametric Maxflows for Structured Sparse Learning with Convex Relaxations of Submodular Functions

TL;DR

This paper introduces a class of structured penalties in convex relaxations of submodular functions that can be efficiently optimized using parametric maxflow algorithms, enabling faster regularized learning.

Contribution

It reveals a broad class of penalties solvable via parametric maxflow, adapting existing algorithms for efficient structured sparse learning with convex relaxations.

Findings

Parametric maxflow algorithms can efficiently solve proximal problems for certain structured penalties.

Several existing penalties meet the conditions for this optimization approach.

Empirical results show improved runtime performance of the proposed framework.

Abstract

The proximal problem for structured penalties obtained via convex relaxations of submodular functions is known to be equivalent to minimizing separable convex functions over the corresponding submodular polyhedra. In this paper, we reveal a comprehensive class of structured penalties for which penalties this problem can be solved via an efficiently solvable class of parametric maxflow optimization. We then show that the parametric maxflow algorithm proposed by Gallo et al. and its variants, which runs, in the worst-case, at the cost of only a constant factor of a single computation of the corresponding maxflow optimization, can be adapted to solve the proximal problems for those penalties. Several existing structured penalties satisfy these conditions; thus, regularized learning with these penalties is solvable quickly using the parametric maxflow algorithm. We also investigate the empirical runtime performance of the proposed framework.