Adversarial robustness is a critical property in a variety of modern machine learning applications. While it has been the subject of several recent theoretical studies, many important questions related to adversarial robustness are still open. In this work, we study a fundamental question regarding Bayes optimality for adversarial robustness. We provide general sufficient conditions under which the existence of a Bayes optimal classifier can be guaranteed for adversarial robustness. Our results can provide a useful tool for a subsequent study of surrogate losses in adversarial robustness and their consistency properties. This manuscript is the extended version of the paper "On the Existence of the Adversarial Bayes Classifier" published in NeurIPS. The results of the original paper did not apply to some non-strictly convex norms. Here we extend our results to all possible norms. Additionally, we clarify a missing step in one of our proofs.

翻译：对抗性稳健性是现代机器各种学习应用中的一个关键属性。 虽然这是最近若干理论研究的主题, 与对抗性稳健性有关的许多重要问题仍然有待解决。 在这项工作中, 我们研究拜斯人的最佳对抗性稳健性的基本问题。 我们为巴耶斯人的最佳分类者的存在提供了一般的足够条件,保证对抗性稳健性。 我们的结果可以为随后研究对抗性稳健性替代损失及其一致性特性提供一个有用的工具。 这本手稿是NeurIPS出版的“关于对抗性海湾分类者的存在”的论文的扩展版。 原始文件的结果没有适用于某些非严格的非约束性规范。 我们在这里将我们的结果扩大到所有可能的规范。 此外, 我们澄清了我们证据中的一个缺失的一步。