Optimal approximation for unconstrained non-submodular minimization

TL;DR

This paper develops the first approximation guarantees for minimizing functions that are nearly submodular, extending submodular minimization techniques to non-submodular functions with theoretical optimality.

Contribution

It introduces a novel framework that extends submodular minimization methods to functions close to submodular, providing the first known optimal approximation guarantees.

Findings

First approximation guarantees for non-submodular minimization.

Guarantees are optimal, matching lower bounds.

Extends to noisy function evaluations.

Abstract

Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function is not exactly submodular, but close. In this case, no theoretical guarantees exist. Indeed, submodular minimization algorithms rely on intricate connections between submodularity and convexity. We show how these relations can be extended to obtain approximation guarantees for minimizing non-submodular functions, characterized by how close the function is to submodular. We also extend this result to noisy function evaluations. Our approximation results are the first for minimizing non-submodular functions, and are optimal, as established by our matching lower bound.