Robust multi-task feature learning

Multi-task learning (MTL) aims to improve the performance of multiple related tasks by exploiting the intrinsic relationships among them. Recently, multi-task feature learning algorithms have received increasing attention and they have been successfully applied to many applications involving high dimensional data. However, they assume that all tasks share a common set of features, which is too restrictive and may not hold in real-world applications, since outlier tasks often exist. In this paper, we propose a Robust Multi-Task Feature Learning algorithm (rMTFL) which simultaneously captures a common set of features among relevant tasks and identifies outlier tasks. Specifically, we decompose the weight (model) matrix for all tasks into two components. We impose the well-known group Lasso penalty on row groups of the first component for capturing the shared features among relevant tasks. To simultaneously identify the outlier tasks, we impose the same group Lasso penalty but on column groups of the second component. We propose to employ the accelerated gradient descent to efficiently solve the optimization problem in rMTFL, and show that the proposed algorithm is scalable to large-size problems. In addition, we provide a detailed theoretical analysis on the proposed rMTFL formulation. Specifically, we present a theoretical bound to measure how well our proposed rMTFL approximates the true evaluation, and provide bounds to measure the error between the estimated weights of rMTFL and the underlying true weights. Moreover, by assuming that the underlying true weights are above the noise level, we present a sound theoretical result to show how to obtain the underlying true shared features and outlier tasks (sparsity patterns). Empirical studies on both synthetic and real-world data demonstrate that our proposed rMTFL is capable of simultaneously capturing shared features among tasks and identifying outlier tasks.