标题： 辅助学习及其统计理解 显示英文标题

标题： Auxiliary Learning and its Statistical Understanding

摘要： 现代统计分析常常遇到高维问题但样本量有限的情况，这对传统统计估计方法提出了巨大挑战。在这项工作中，我们采用辅助学习来解决高维设置下的估计问题。我们从线性回归设定开始。为了提高主要任务参数估计的统计效率，我们考虑了几个辅助任务，这些辅助任务与主要任务共享相同的协变量。然后，我们开发了一个主要任务的加权估计器，它是主要任务和辅助任务的普通最小二乘估计器的线性组合。最优权重被解析地推导出来，并研究了相应加权估计器的统计性质。随后，我们将加权估计器扩展到广义线性回归模型。进行了大量的数值实验以验证我们的理论结果。最后，为了说明目的，给出了一个与智能自动售货机相关的深度学习实际数据例子。 显示英文摘要

摘要： Modern statistical analysis often encounters high-dimensional problems but with a limited sample size. It poses great challenges to traditional statistical estimation methods. In this work, we adopt auxiliary learning to solve the estimation problem in high-dimensional settings. We start with the linear regression setup. To improve the statistical efficiency of the parameter estimator for the primary task, we consider several auxiliary tasks, which share the same covariates with the primary task. Then a weighted estimator for the primary task is developed, which is a linear combination of the ordinary least squares estimators of both the primary task and auxiliary tasks. The optimal weight is analytically derived and the statistical properties of the corresponding weighted estimator are studied. We then extend the weighted estimator to generalized linear regression models. Extensive numerical experiments are conducted to verify our theoretical results. Last, a deep learning-related real-data example of smart vending machines is presented for illustration purposes.