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We consider the problem of learning a structured multi-task regression, wherethe output consists of multiple responses that are related by a graph and thecorrelated response variables are dependent on the common inputs in a sparsebut synergistic manner. Previous methods such as l1/l2-regularized multi-taskregression assume that all of the output variables are equally related to theinputs, although in many real-world problems, outputs are related in a complexmanner. In this paper, we propose graph-guided fused lasso (GFlasso) forstructured multi-task regression that exploits the graph structure over theoutput variables. We introduce a novel penalty function based on fusion penaltyto encourage highly correlated outputs to share a common set of relevantinputs. In addition, we propose a simple yet efficient proximal-gradient methodfor optimizing GFlasso that can also be applied to any optimization problemswith a convex smooth loss and the general class of fusion penalty defined onarbitrary graph structures. By exploiting the structure of the non-smooth''fusion penalty'', our method achieves a faster convergence rate than thestandard first-order method, sub-gradient method, and is significantly morescalable than the widely adopted second-order cone-programming andquadratic-programming formulations. In addition, we provide an analysis of theconsistency property of the GFlasso model. Experimental results not onlydemonstrate the superiority of GFlasso over the standard lasso but also showthe efficiency and scalability of our proximal-gradient method.

Graph-Structured Multi-task Regression and an Efficient Optimization Method for General Fused Lasso