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.