CAREER: Adversarial Robustness through the Lens of Mathematical Analysis and Geometry

过去几年，人工智能和机器学习以闪电般的速度在多个应用领域得到了扩展。这种前所未有的技术发展激发了新的范例，在安全关键应用中至关重要，用于判断数据分析方法的成功，取代高预测能力作为训练模型的唯一标准，并为更细致入微的学习提供更多相关性，可以考虑可靠性，隐私性和公平性标准。该项目将探索基础问题，理论和算法，与范式的扩展相一致，并为本科生和研究生提供研究机会。该项目在分析和几何特征方面与众不同，旨在为机器学习中的及时问题提供一个总体视角，这些问题将触发对抗性鲁棒性和隐私以及最优运输、逆问题、相互作用粒子系统和平均场偏微分方程、几何类型偏微分方程分析和几何变分问题等领域之间的各种联系。本项目将追求的一些具体目标包括：分析各种学习环境下对抗性攻击的几何结构，了解对抗性训练的正则化能力，研究对抗性训练与不同形式的显式正则化方法之间的联系，开发算法策略，利用度量空间中的不同几何结构来增强鲁棒性和隐私性，设计和研究新的对抗性训练几何框架。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The past few years have witnessed an expansion of artificial intelligence and machine learning in several domains of applications at lightning speed. This unprecedented technological development has motivated new paradigms, essential in safety critical applications, for judging the success of data analysis methodologies, displacing high predictive power as the sole criterion for training models and giving more relevance to a more nuanced learning that can factor in reliability, privacy, and fairness criteria. This project will explore foundational questions, theoretical and algorithmic, in line with this expansion of paradigms, and provide research opportunities for undergraduate and graduate students.This project is distinctive in its analytical and geometric character and aims at providing an overarching perspective on timely problems in machine learning that will trigger a variety of connections between adversarial robustness and privacy and fields like optimal transportation, inverse problems, interacting particle systems and mean field PDEs, analysis of PDEs of geometric type, and geometric variational problems, among others. Some concrete goals that will be pursued in this project include to: analyze the geometric structure of adversarial attacks in a variety of learning settings, understand the regularization capabilities of adversarial training, study the connections between adversarial training and different forms of explicit regularization methods, develop algorithmic strategies, by exploiting different geometric structures in spaces of measures, to enforce robustness and privacy, and design and study of new geometric frameworks for adversarial training.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.