  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.
