The reliability of a learning model is key to the successful deployment of machine learning in various industries. Creating a robust model, particularly one unaffected by adversarial attacks, requires a comprehensive understanding of the adversarial examples phenomenon. However, it is difficult to describe the phenomenon due to the complicated nature of the problems in machine learning. Consequently, many studies investigate the phenomenon by proposing a simplified model of how adversarial examples occur and validate it by predicting some aspect of the phenomenon. While these studies cover many different characteristics of the adversarial examples, they have not reached a holistic approach to the geometric and analytic modeling of the phenomenon. This paper propose a formal framework to study the phenomenon in learning theory and make use of complex analysis and holomorphicity to offer a robust learning rule for artificial neural networks. With the help of complex analysis, we can effortlessly move between geometric and analytic perspectives of the phenomenon and offer further insights on the phenomenon by revealing its connection with harmonic functions. Using our model, we can explain some of the most intriguing characteristics of adversarial examples, including transferability of adversarial examples, and pave the way for novel approaches to mitigate the effects of the phenomenon.

该研究论文提出了一种正式框架以研究机器学习中敌对示例现象，并利用复分析和全纯性提出了一种针对人工神经网络的稳健学习规则，揭示了该现象与调和函数的联系，并能够解释敌对示例的许多特征，包括可转移性，并为缓解敌对示例的影响铺平道路。

人工神经网络稳健训练的分析框架