Collaborative Research: High-Dimensional Decision Making and Inference with Applications for Personalized Medicine

随着数据采集和存储技术的出现，研究人员可以以较低的价格获得大规模、高维的数据集。这些数据集提供了令人兴奋的机会，可以做出更好的决策，揭示新的发现，以改善各种应用中的决策，同时也提出了统计挑战。在过去的几十年里，已经提出了正则化方法，如Lasso， SCAD和MCP，用于在存在高维协变量的情况下进行模型估计。针对这些方法开发了各种数值算法，并对它们的理论性质进行了很好的研究。然而，如何高效、有效地利用高维数据进行最优决策和推理的研究相对较少，尽管这些问题具有重要的现实意义。本项目将发展在高维环境下做出最优决策和进行有效推理的新方法和理论。这些方法有广泛的应用，例如，在个性化医疗中，目标是根据预测信息（包括数千个遗传标记）确定患者的最佳治疗方法。主要研究人员将开发和分发用户友好的开源软件给实践者，并为不同层次的学生提供培训机会。该项目有三个研究目标。第一个目标是研究具有二元动作的高维上下文强盗问题，这是一个在个性化医疗和精准医疗中应用的在线决策问题。在这个问题中，玩家依次选择一个行动并观察奖励，其目标是最大化奖励。主要研究人员将开发一种新的算法来提供最优决策规则，以实现最小最大最优后悔。第二个目标是研究由高维随机凸优化引起的一般推理问题，其目标是量化最优目标值的不确定性。第三个目标是考虑具有有限随机行动空间的一般随机线性强盗问题。研究人员将利用最佳子集选择型估计器开发一种新的算法，该方法在低维设置下实现了“无维”遗憾，并满足现有的下界。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。

With the advent of data collection and storage technology, researchers can obtain large-scale and high-dimensional datasets at a low price. Such datasets offer exciting opportunities to make better decisions and reveal new discoveries to improve decision making in various applications, and meanwhile, also raise statistical challenges. Over the past decades, regularization methods such as Lasso, SCAD, and MCP have been proposed to conduct model estimation in the presence of high dimensional covariates. Various numerical algorithms have been developed for these methods, and their theoretical properties are well studied. However, questions of how to efficiently and effectively utilize high-dimensional data to make optimal decisions and conduct inference are relatively less studied, although such problems are of vital practical importance. This project will develop new methods and theories for making optimal decisions and conducting valid inference under high-dimensional settings. The methods have wide applications, for instance, in personalized medicine where the goal is to determine the optimal treatments for a patient based on predictor information, including several thousand genetic markers. The principal investigators will develop and distribute user-friendly open-source software to practitioners and provide training opportunities to students at different levels. The project has three research aims. The first aim is to study the high-dimensional contextual bandit problem with binary actions, which is an online decision-making problem that finds applications in personalized healthcare and precision medicine. In this problem, the player sequentially chooses one action and observes a reward, where the goal is to maximize the reward. The principal investigators will develop a new algorithm to provide an optimal decision rule, which achieves the minimax optimal regret. The second aim is to study general inference problems that arise from high-dimensional stochastic convex optimization, where the goal is to quantify the uncertainties of the optimal objective value. The third goal is to consider the general stochastic linear bandit problem with a finite and random action space. The principal investigators will develop a new algorithm by using a best-subset-selection type estimator, and the approach achieves a "dimension-free" regret and meets existing lower-bound under the low-dimensional setting.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.