  The group Lasso is an extension of the Lasso for feature selection on
(predefined) non-overlapping groups of features. The non-overlapping group
structure limits its applicability in practice. There have been several recent
attempts to study a more general formulation, where groups of features are
given, potentially with overlaps between the groups. The resulting optimization
is, however, much more challenging to solve due to the group overlaps. In this
paper, we consider the efficient optimization of the overlapping group Lasso
penalized problem. We reveal several key properties of the proximal operator
associated with the overlapping group Lasso, and compute the proximal operator
by solving the smooth and convex dual problem, which allows the use of the
gradient descent type of algorithms for the optimization. We have performed
empirical evaluations using the breast cancer gene expression data set, which
consists of 8,141 genes organized into (overlapping) gene sets. Experimental
results demonstrate the efficiency and effectiveness of the proposed algorithm.
