Submodular Minimax Optimization: Finding Effective Sets

TL;DR

This paper introduces a framework for submodular minimax optimization in combinatorial settings, providing theoretical characterizations and demonstrating its effectiveness in various machine learning applications.

Contribution

It offers the first comprehensive characterization of submodular minimax optimization and develops algorithms that outperform baselines in multiple practical tasks.

Findings

Algorithms outperform baselines in experiments

Provides theoretical conditions for effective set finding

Demonstrates robustness in machine learning applications

Abstract

Despite the rich existing literature about minimax optimization in continuous settings, only very partial results of this kind have been obtained for combinatorial settings. In this paper, we fill this gap by providing a characterization of submodular minimax optimization, the problem of finding a set (for either the min or the max player) that is effective against every possible response. We show when and under what conditions we can find such sets. We also demonstrate how minimax submodular optimization provides robust solutions for downstream machine learning applications such as (i) efficient prompt engineering for question answering, (ii) prompt engineering for dialog state tracking, (iii) identifying robust waiting locations for ride-sharing, (iv) ride-share difficulty kernelization, and (v) finding adversarial images. Our experiments demonstrate that our proposed algorithms consistently outperform other baselines.