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.

翻译：尽管连续空间中的最小最大优化已有丰富文献，但关于组合环境的最小最大优化仅有部分零散结果。本文通过刻画子模最小最大优化问题填补了这一空白，该问题旨在寻找（对最小化或最大化玩家而言）能有效应对所有可能响应的集合。我们阐明了在何种条件下可以找到此类集合，并展示了子模最小最大优化如何为下游机器学习应用提供鲁棒解决方案，包括：(i) 问答系统的提示词高效工程构建，(ii) 对话状态跟踪的提示词工程，(iii) 网约车鲁棒候车点识别，(iv) 网约车难度核化，以及(v) 对抗性图像生成。实验表明，我们提出的算法始终优于其他基线方法。