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.