CAREER: Smooth statistical distances for a scalable learning theory

机器学习是许多面向人类的应用的支柱，从自动驾驶汽车到医疗诊断。尽管取得了广泛的经验性进展，但目前的理论智慧福尔斯未能提供有意义的性能保证，以解决与此类技术相关的安全问题。差距在操作真实世界的高维数据的系统中尤其明显，这是最需要理论保证的地方。该项目将开发一种新的高维推理框架，从而对现代机器学习方法进行可扩展的统计分析。这项创新将允许耦合经验验证与原则性的性能评估技术和可证明的准确性保证。最终，该项目将促进机器学习技术的广泛部署，带来宝贵的社会效益，从更好的医疗保健到更安全的道路和更好的危机管理。同时，教育部分将培养理论STEM学科的下一代科学家，同时增加妇女和女孩的参与，她们在很大程度上仍然代表性不足。该项目引入了平滑统计距离--一种新的适应高维空间的概率分布之间的差异度量。平滑距离消除了测量分布中的局部不规则性（通过与所选内核的卷积），从而保留了推理能力，但在从数据估计这些距离时消除了维数灾难。由于测量或优化分布之间的距离是基本推理设置和高级机器学习任务的核心，因此研究议程包括三个时间顺序阶段：（1）开发平滑距离的基础知识，包括几何，拓扑和功能属性;（2）进行高维统计研究，从经验近似问题到基本推理;以及（3）利用前两项任务中获得的知识，为机器学习任务（如生成建模、重心计算和信息流分析）设计一个精细的泛化和样本复杂性理论。平滑统计距离范式有可能弥合中心理论差距，并在大规模机器学习接口中提供更高的可靠性。该奖项反映了NSF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。

Machine learning forms the backbone of many human-facing applications, from autonomous vehicles to medical diagnostics. Despite extensive empirical progress, current theoretical wisdom falls short of providing meaningful performance guarantees that address safety ramifications associated with such technologies. The gap is especially pronounced in systems that operate on real-world, high-dimensional data, which is where theoretical guarantees are most needed. This project will develop a novel framework for high-dimensional inference that gives rise to a scalable statistical analysis of modern machine learning methods. This innovation will allow to couple empirical validation with principled performance evaluation techniques and provable accuracy assurances. Ultimately, this project will promote the wide deployment of machine learning technologies with invaluable societal benefits, from better healthcare to safer roads and improved crisis management. In conjunction, the educational component will nurture the next generation of scientists in theoretical STEM disciplines, while increasing participation of women and girls, who remain largely underrepresented.This project introduces smooth statistical distances---a new class of discrepancy measures between probability distributions adapted to high-dimensional spaces. Smooth distances level out local irregularities in the measured distributions (via convolution with a chosen kernel) in a way that preserves inference capability, but alleviates the curse of dimensionality when estimating these distances from data. Since measuring or optimizing distances between distributions is central to basic inference setups and advanced machine learning tasks, the research agenda comprises three chronological phases: (1) develop fundamentals of smooth distances, encompassing geometric, topological and functional properties; (2) conduct a high-dimensional statistical study, from empirical approximation questions to basic inference; and (3) devise a refined generalization and sample complexity theory for machine learning tasks like generative modeling, barycenter computation, and information flow analysis, drawing on knowledge gained during the first two tasks. The smooth statistical distances paradigm has the potential to bridge central theoretical gaps and provide increased reliability in machine learning interfaces at scale.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.