  The sparse group lasso optimization problem is solved using a coordinate
gradient descent algorithm. The algorithm is applicable to a broad class of
convex loss functions. Convergence of the algorithm is established, and the
algorithm is used to investigate the performance of the multinomial sparse
group lasso classifier. On three different real data examples the multinomial
group lasso clearly outperforms multinomial lasso in terms of achieved
classification error rate and in terms of including fewer features for the
classification. The run-time of our sparse group lasso implementation is of the
same order of magnitude as the multinomial lasso algorithm implemented in the R
package glmnet. Our implementation scales well with the problem size. One of
the high dimensional examples considered is a 50 class classification problem
with 10k features, which amounts to estimating 500k parameters. The
implementation is available as the R package msgl.
