  Sparse learning has recently received increasing attention in many areas
including machine learning, statistics, and applied mathematics. The mixed-norm
regularization based on the l1q norm with q>1 is attractive in many
applications of regression and classification in that it facilitates group
sparsity in the model. The resulting optimization problem is, however,
challenging to solve due to the inherent structure of the mixed-norm
regularization. Existing work deals with special cases with q=1, 2, infinity,
and they cannot be easily extended to the general case. In this paper, we
propose an efficient algorithm based on the accelerated gradient method for
solving the general l1q-regularized problem. One key building block of the
proposed algorithm is the l1q-regularized Euclidean projection (EP_1q). Our
theoretical analysis reveals the key properties of EP_1q and illustrates why
EP_1q for the general q is significantly more challenging to solve than the
special cases. Based on our theoretical analysis, we develop an efficient
algorithm for EP_1q by solving two zero finding problems. To further improve
the efficiency of solving large dimensional mixed-norm regularized problems, we
propose a screening method which is able to quickly identify the inactive
groups, i.e., groups that have 0 components in the solution. This may lead to
substantial reduction in the number of groups to be entered to the
optimization. An appealing feature of our screening method is that the data set
needs to be scanned only once to run the screening. Compared to that of solving
the mixed-norm regularized problems, the computational cost of our screening
test is negligible. The key of the proposed screening method is an accurate
sensitivity analysis of the dual optimal solution when the regularization
parameter varies. Experimental results demonstrate the efficiency of the
proposed algorithm.
