Correlated outcomes are common in many practical problems. In some settings, one outcome is of particular interest, and others are auxiliary. To leverage information shared by all the outcomes, traditional multi-task learning (MTL) minimizes an averaged loss function over all the outcomes, which may lead to biased estimation for the target outcome, especially when the MTL model is mis-specified. In this work, based on a decomposition of estimation bias into two types, within-subspace and against-subspace, we develop a robust transfer learning approach to estimating a high-dimensional linear decision rule for the outcome of interest with the presence of auxiliary outcomes. The proposed method includes an MTL step using all outcomes to gain efficiency, and a subsequent calibration step using only the outcome of interest to correct both types of biases. We show that the final estimator can achieve a lower estimation error than the one using only the single outcome of interest. Simulations and real data analysis are conducted to justify the superiority of the proposed method.

翻译：在许多实际问题中，相关结果变量普遍存在。在某些场景下，某一结果变量具有特定研究意义，其余结果则作为辅助变量。为利用所有结果变量共享的信息，传统多任务学习（MTL）通过最小化所有结果变量的平均损失函数进行建模，但这可能导致目标结果变量的估计产生偏差，尤其当MTL模型设定错误时。本研究基于对估计偏差的两种类型——子空间内偏差与子空间外偏差——的分解，提出一种稳健的迁移学习方法，用于在存在辅助结果变量的情况下估计目标结果的高维线性决策规则。该方法首先利用所有结果变量进行MTL步骤以提升效率，随后仅使用目标结果变量进行校准步骤，以纠正上述两类偏差。理论证明表明，最终估计量可实现比仅使用单一目标结果变量时更低的估计误差。通过模拟实验与真实数据分析验证了所提方法的优越性。