  Multi-task learning (MTL) aims to improve estimation and prediction
performance by sharing common information among related tasks. One natural
assumption in MTL is that tasks are classified into clusters based on their
characteristics. However, existing MTL methods based on this assumption often
ignore outlier tasks that have large task-specific components or no relation to
other tasks. To address this issue, we propose a novel MTL method called
Multi-Task Learning via Robust Regularized Clustering (MTLRRC). MTLRRC
incorporates robust regularization terms inspired by robust convex clustering,
which is further extended to handle non-convex and group-sparse penalties. The
extension allows MTLRRC to simultaneously perform robust task clustering and
outlier task detection. The connection between the extended robust clustering
and the multivariate M-estimator is also established. This provides an
interpretation of the robustness of MTLRRC against outlier tasks. An efficient
algorithm based on a modified alternating direction method of multipliers is
developed for the estimation of the parameters. The effectiveness of MTLRRC is
demonstrated through simulation studies and application to real data.
