Robust Wasserstein Profile Inference

在不确定的情况下，任何数据驱动的决策问题的一个关键部分涉及能够以高度的置信度保证估计的决策规则在实际应用时具有理想的性能。准确地说，这是统计推断所提供的保证的关键作用。这个项目的目标是研究一种新的推理方法，这种方法从一开始就精确地构建具有增强的样本外属性的数据驱动的决策规则。这是通过引入博弈论公式来实现的，在该公式中，决策者针对对手进行优化，该对手在向数据添加扰动(在合理大小但任意方向内)时最佳地利用决策的潜在弱点。设计了一个统计框架来估计最优量的数据扰动，以获得稳健而实用的决策规则。该框架自然会导致潜在决策参数的最优(在某种意义上)置信域。该项目的成果在不确定情况下的应用决策的各个领域都有影响，特别是机器学习、人工智能、运营管理和交通运输是特别感兴趣的应用。研究生将致力于大规模最优运输的计算方法。本项目要研究的新的推理方法统一和扩展了一大类估计量(如广义Lasso和正则化Logistic回归等)，并已在实践中成功应用。这些都包含在分布式稳健优化(DRO)框架中。DRO公式是一类博弈，在这类博弈中，统计学家选择一个参数或行动来最小化某些预期损失，而对手选择在一定大小(称为分布不确定性大小)内对统计学家(最大化预期损失)进行经验度量的扰动。在这一提议的背景下，这种扰动是根据最优运输成本(或瓦瑟斯坦距离)来衡量的。沃瑟斯坦距离和DRO公式的使用证明了这种推理方法的稳健沃瑟斯坦轮廓推断的名称是正确的。该方案研究了由DRO公式引起的分布不确定性大小和相关最优置信域的最优选择。具体的应用，例如，形状约束估计和工程中的随机优化问题，将由PI来研究。所提出的研究提供了最优运输理论、统计推理和凸优化之间的丰富的相互作用。最后，PI将试图从代表性不足的群体中招聘高素质的人员。除了标准的工具(会议和期刊出版物)外，PI还将通过开放获取网站传播这项提案的科学成果。这一奖项反映了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。

A crucial part of any data-driven decision-making problem under uncertainty involves being able to guarantee, with a high degree of confidence, a desirable performance of estimated decision rules when actually deployed in practice. This, precisely, is the key role of the guarantees given by statistical inference. The goal of this project is to investigate a novel inference methodology that precisely builds from inception data-driven decision-making rules with enhanced out-of-sample properties. This is achieved by introducing a game-theoretic formulation, in which the decision maker optimizes against an adversary that optimally exploits potential weaknesses of a decision when adding perturbations to the data (within reasonable size, yet arbitrary directions). A statistical framework is designed to estimate an optimal amount of data perturbations to obtain robust, yet practical, decision rules. The framework naturally leads to optimal (in certain sense) confidence regions for the underlying decision making parameters. The output of this project has implications in various areas of applied decision making under uncertainty, in particular, machine learning, artificial intelligence, operations management, and transportation are applications of special interest. The graduate student will work on computational methods for large scale optimal transport. The novel inference methodology to be investigated in this project unifies and extends a large class of estimators (such as generalized Lasso and regularized logistic regression among many others), which have been successfully applied in practice. These are encompassed within a distributionally robust optimization (DRO) framework. A DRO formulation is a class of games in which the statistician chooses a parameter or an action to minimize certain expected loss and an adversary chooses a perturbation of the empirical measure against the statistician (maximizing the expected loss) within a certain size (called the distributional uncertainty size). In the context of this proposal, this perturbation is measured in terms of optimal transport costs (or Wasserstein distances). The use of the Wasserstein distance and the DRO formulation justifies the name Robust Wasserstein Profile Inference of this inference methodology. The proposal studies the optimal selection of the distributional uncertainty size and associated optimal confidence regions induced by the DRO formulation. Specific applications, for example, to shape constrained estimation and stochastic optimization problem in engineering will be studied by the PI. The proposed research provides a rich interplay between the theory of optimal transport, statistical inference and convex optimization. Finally, the PI will attempt to recruit high-quality personnel from under-represented groups. The PI will also disseminate the scientific output of this proposal via open access sites, in addition to the standard vehicles (conferences and journal publications).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.