Multi-Stage Multi-Task Learning with Reduced Rank

Multi-task learning (MTL) seeks to improve the generalization performance by sharing information among multiple tasks. Many existing MTL approaches aim to learn the lowrank structure on the weight matrix, which stores the model parameters of all tasks, to achieve task sharing, and as a consequence the trace norm regularization is widely used in the MTL literature. A major limitation of these approaches based on trace norm regularization is that all the singular values of the weight matrix are penalized simultaneously, leading to impaired estimation on recovering the larger singular values in the weight matrix. To address the issue, we propose a Reduced rAnk MUlti-Stage multi-tAsk learning (RAMUSA) method based on the recently proposed capped norms. Different from existing trace-norm-based MTL approaches which minimize the sum of all the singular values, the RAMUSA method uses a capped trace norm regularizer to minimize only the singular values smaller than some threshold. Due to the non-convexity of the capped trace norm, we develop a simple but well guaranteed multi-stage algorithm to learn the weight matrix iteratively. We theoretically prove that the estimation error at each stage in the proposed algorithm shrinks and finally achieves a lower upper-bound as the number of stages becomes large enough. Empirical studies on synthetic and real-world datasets demonstrate the effectiveness of the RAMUSA method in comparison with the state-of-the-art methods.