  The problem of joint feature selection across a group of related tasks has
applications in many areas including biomedical informatics and computer
vision. We consider the l2,1-norm regularized regression model for joint
feature selection from multiple tasks, which can be derived in the
probabilistic framework by assuming a suitable prior from the exponential
family. One appealing feature of the l2,1-norm regularization is that it
encourages multiple predictors to share similar sparsity patterns. However, the
resulting optimization problem is challenging to solve due to the
non-smoothness of the l2,1-norm regularization. In this paper, we propose to
accelerate the computation by reformulating it as two equivalent smooth convex
optimization problems which are then solved via the Nesterov's method-an
optimal first-order black-box method for smooth convex optimization. A key
building block in solving the reformulations is the Euclidean projection. We
show that the Euclidean projection for the first reformulation can be
analytically computed, while the Euclidean projection for the second one can be
computed in linear time. Empirical evaluations on several data sets verify the
efficiency of the proposed algorithms.
