Robust Uncertainty Quantification and Statistical Learning for Heavy Tails and Rare Events

基于概率和统计的数学模型被广泛用于各种环境中，例如在人工智能、金融和运筹学、物理科学和许多其他领域。人们感兴趣的一个核心问题是，可以在多大程度上信任给定模型的预测。正如经常发生的那样，当模型本身的性质存在重大不确定性时，这一点至关重要。在这个项目中，研究人员的目标是开发一个基于信息论的系统数学框架来解决这些问题。这项研究的重点之一将是预测罕见(但可能是灾难性的)事件的概率。例如，如果一个给定的模型预测一个灾难性事件是一个百年一遇的事件，该模型中的不确定性如何潜在地改变这一预测？调查人员将建立相应的压力测试，以评估不确定性的影响。这项研究还将提供系统的工具，用数据训练新的统计学习模型，并提供性能保证。这项研究将集中在复杂系统不确定性量化的概率基础的发展和统计学习的相关问题上。总体目标是在模型本身存在不确定性时提供可计算的性能保证，以及开发可信和可靠的推理算法。许多不同的指标和信息理论测量可用于比较概率分布(例如，库尔巴克-莱布勒散度)；该项目的一个统一主题是以原则性的方式确定哪种方法最适合特定任务。在这种情况下，任务包括通过评估某些感兴趣的量从模型中提取信息，例如平均值、方差、某些罕见事件的概率等。利用变分原理，将推导出新的最优信息不等式来解决这些问题。从稳健性的角度来看，这允许设计微调的压力测试，即根据金融机构使用的压力测试的精神，围绕给定的基线模型构建模型社区并计算最糟糕的情况，以防止在替代方案下发生突然变化。在统计学习，特别是近似推理的背景下，中心挑战是1)选择适当的发散最小化作为学习概率模型的手段，以及2)为学习过程提供性能保证。调查人员将重点研究这些问题，重点放在感兴趣的数量很少(但可能是灾难性的)事件的情况下。他们将评估模型不确定性对这些灾难性事件和重尾模型的影响。该项目旨在为概率算法的性能保证提供数学基础，这些算法用于从材料科学到运筹学、机器学习和人工智能的广泛问题。专注于对极端和罕见事件的可靠预测，使该项目具有及时性和广泛的适用性。例如，稳健的不确定性量化视角为安全关键问题(如海洋中的无赖海浪或电网故障)提供了最坏情况的解决方案、压力测试和偏差控制。此外，近似推理的概率性能保证可以使现有的黑盒推理算法以数学系统的方式更加可信和透明。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical models based on probability and statistics are used in a wide variety of contexts, for example in artificial intelligence, finance and operations research, physical sciences, and many others. A central question of interest is how much trust can be put in the predictions of a given model. This is crucial when, as it often happens, there are significant uncertainties associated with the nature of the model itself. In this project, the investigators aim to develop a systematic mathematical framework based on the theory of information to address these issues. One focus of the research will be on prediction of the probability of rare (but potentially catastrophic) events. For example, if a given model predicts a catastrophic event to be a 100-year event, how do uncertainties in the model potentially change this prediction? The investigators will build corresponding stress tests to assess the effects of uncertainties. The research will also provide systematic tools to train new statistical learning models with data and provide performance guarantees. This research will focus on the development of the probabilistic foundations of uncertainty quantification for complex systems and on related questions about statistical learning. The overarching goals are to provide computable performance guarantees when there is uncertainty in the model itself as well as to develop trustworthy and reliable inference algorithms. Many different metrics and information theoretic measurements are available to compare probability distributions (for example, the Kullback-Leibler divergence); a unifying theme of the project is to determine, in a principled manner, which method is most appropriate to a specific task. In this context, a task consists of extracting information from the model by evaluating certain quantities of interest, such as average values, variance, probability of some rare event, and so on. Using variational principles, new optimal information inequalities will be derived to address these issues. From a robustness perspective, this allows the design of finely tuned stress tests, that is, to build neighborhoods of models around a given baseline model and to compute worst-case scenarios, in the spirit of the stress tests used by financial institutions to protect against sudden changes under alternative scenarios. In the context of statistical learning, and especially approximate inference, central challenges are 1) to select the right divergence to minimize as means to learn probabilistic models, and 2) to provide performance guarantees for the learning process. The investigators will study these questions with emphasis on the case where the quantities of interest are rare (but potentially catastrophic) events. They will assess the impact of model uncertainty on these catastrophic events and on models with heavy tails. The project aims to provide mathematical foundations for performance guarantees in probabilistic algorithms used in a wide array of problems from materials science, to operations research, machine learning, and artificial intelligence. The focus on reliable predictions of extreme and rare events makes the project timely and widely applicable. For example, the robust uncertainty quantification perspective provides worst-case solutions, stress tests, and bias control for safety-critical problems (such as rogue waves in the ocean or power grid failure). Furthermore, probabilistic performance guarantees for approximate inference can make existing black box inference algorithms more trustworthy and transparent in a mathematically systematic manner.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.