We consider the problem of learning a sparse multi-task regression with an application to a genetic association mapping problem for discovering genetic markers that influence expression levels of multiple genes jointly. In particular, we consider the case where the structure over the outputs can be represented as a tree with leaf nodes as outputs and internal nodes as clusters of the outputs at multiple granularity, and aim to recover the common set of relevant inputs for each output cluster. Assuming that the tree structure is available as a prior knowledge, we formulate this problem as a new multi-task regularized regression called tree-guided group lasso. Our structured regularization is based on a group-lasso penalty, where the group is 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 optimization method that can handle a large-scale problem as is typically the case in association mapping that involve thousands of genes as outputs and millions of genetic markers as inputs. 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.
REF proposed a tree-guided group lasso, or tree lasso, that directly combined statistical strength across multiple related outputs.
Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping
Annals of Applied Statistics 2012, Vol. 6, No. 3, 1095-1117
Mathematics Biology