Learning tree structure in multi-task learning

In multi-task learning (MTL), multiple related tasks are learned jointly by sharing information according to task relations. One promising approach is to utilize the given tree structure, which describes the hierarchical relations among tasks, to learn model parameters under the regularization framework. However, such a priori information is rarely available in most applications. To the best of our knowledge, there is no work to learn the tree structure among tasks and model parameters simultaneously under the regularization framework and in this paper, we develop a TAsk Tree (TAT) model for MTL to achieve this. By specifying the number of layers in the tree as H, the TAT method decomposes the parameter matrix into H component matrices, each of which corresponds to the model parameters in each layer of the tree. In order to learn the tree structure, we devise sequential constraints to make the distance between the parameters in the component matrices corresponding to each pair of tasks decrease over layers, and hence the component parameters will keep fused until the topmost layer, once they become fused in a layer. Moreover, to make the component parameters have chance to fuse in different layers, we develop a structural sparsity regularizer, which is the sum of the ℓ₂ norm on the pairwise difference among the component parameters, to learn layer-specific task structure. In order to solve the resulting non-convex objective function, we use the general iterative shrinkage and thresholding (GIST) method. By using the alternating direction method of multipliers (ADMM) method, we decompose the proximal problem in the GIST method into three independent subproblems, where a key subproblem with the sequential constraints has an efficient solution as the other two subproblems do. We also provide some theoretical analysis for the TAT model. Experiments on both synthetic and real-world datasets show the effectiveness of the TAT model.