Prismatic Algorithm for Discrete D.C. Programming Problems

TL;DR

This paper introduces the first exact branch-and-bound algorithm for minimizing the difference of two submodular functions, enabling precise solutions for complex set function optimization problems in machine learning.

Contribution

It presents a novel exact algorithm for discrete D.C. programming problems, leveraging submodularity and convexity relationships, with empirical validation.

Findings

The algorithm effectively solves D.S. problems exactly.

It outperforms existing approximate methods in feature selection.

Exact solutions improve learning tasks accuracy.

Abstract

In this paper, we propose the first exact algorithm for minimizing the difference of two submodular functions (D.S.), i.e., the discrete version of the D.C. programming problem. The developed algorithm is a branch-and-bound-based algorithm which responds to the structure of this problem through the relationship between submodularity and convexity. The D.S. programming problem covers a broad range of applications in machine learning because this generalizes the optimization of a wide class of set functions. We empirically investigate the performance of our algorithm, and illustrate the difference between exact and approximate solutions respectively obtained by the proposed and existing algorithms in feature selection and discriminative structure learning.