Tree-guided group Lasso for multi-task regression with structured sparsity

We consider the problem of learning a sparse multi-task regression, where the structure in the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granular-ity. Our goal is to recover the common set of relevant inputs for each output cluster. Assuming that the tree structure is avail-able as prior knowledge, we formulate this problem as a new multi-task regularized re-gression called tree-guided group lasso. Our structured regularization is based on a group-lasso penalty, where groups are defined with respect to the tree structure. We describe a systematic weighting scheme for the groups in the penalty such that each output variable is penalized in a balanced manner even if the groups overlap. We present an efficient op-timization method that can handle a large-scale problem. Using simulated and yeast datasets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns compared to other methods for multi-task learning. 1