Integrating low-rank and group-sparse structures for robust multi-task learning

Multi-task learning (MTL) aims at improving the generalization performance by utilizing the intrinsic relationships among multiple related tasks. A key assumption in most MTL algorithms is that all tasks are related, which, however, may not be the case in many real-world applications. In this paper, we propose a robust multi-task learning (RMTL) algorithm which learns multiple tasks simultaneously as well as identifies the irrelevant (outlier) tasks. Specifically, the proposed RMTL algorithm captures the task relationships using a low-rank structure, and simultaneously identifies the outlier tasks using a group-sparse structure. The proposed RMTL algorithm is formulated as a non-smooth convex (unconstrained) optimization problem. We propose to adopt the accelerated proximal method (APM) for solving such an optimization problem. The key component in APM is the computation of the proximal operator, which can be shown to admit an analytic solution. We also theoretically analyze the effectiveness of the RMTL algorithm. In particular, we derive a key property of the optimal solution to RMTL; moreover, based on this key property, we establish a theoretical bound for characterizing the learning performance of RMTL. Our experimental results on benchmark data sets demonstrate the effectiveness and efficiency of the proposed algorithm.