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

重尾和稀有事件的鲁棒不确定性量化和统计学习

基本信息

批准号：
2008970
负责人：
Luc Rey-Bellet
金额：
$ 37万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2008970&HistoricalAwards=false
关键词：
Robust Uncertainty Quantification Statistical Learning

项目摘要

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.

基于概率和统计的数学模型广泛应用于各种领域，例如人工智能、金融和运筹学、物理科学等。一个令人感兴趣的中心问题是对给定模型的预测可以有多少信任。当模型本身的性质存在重大不确定性时（这种情况经常发生），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。该研究的重点之一是预测罕见（但可能是灾难性）事件的概率。例如，如果给定模型预测灾难性事件将持续 100 年，那么模型中的不确定性如何可能改变此预测？研究人员将建立相应的压力测试来评估不确定性的影响。该研究还将提供系统工具，用数据训练新的统计学习模型并提供性能保证。这项研究将重点关注复杂系统不确定性量化的概率基础的发展以及统计学习的相关问题。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，以及开发值得信赖和可靠的推理算法。许多不同的度量和信息论测量可用于比较概率分布（例如，Kullback-Leibler 散度）；该项目的一个统一主题是原则性地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（例如平均值、方差、某些罕见事件的概率等）来从模型中提取信息。使用变分原理，将导出新的最优信息不平等来解决这些问题。从稳健性的角度来看，这允许设计精细调整的压力测试，即围绕给定的基线模型构建模型邻域并计算最坏的情况，本着金融机构用于防止替代场景下突然变化的压力测试的精神。在统计学习（尤其是近似推理）的背景下，主要挑战是 1）选择正确的散度以最小化作为学习概率模型的手段，2）为学习过程提供性能保证。研究人员将研究这些问题，重点关注感兴趣的数量是罕见（但可能是灾难性）事件的情况。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，该算法用于解决从材料科学到运筹学、机器学习和人工智能等各种问题。对极端和罕见事件的可靠预测的关注使得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（例如海洋中的狂浪或电网故障）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值得信赖和透明。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。