A note on the group lasso and a sparse group lasso

TL;DR

This paper explores a generalized sparse group lasso penalty that combines lasso and group lasso, providing an efficient coordinate descent algorithm applicable to non-orthonormal model matrices.

Contribution

It introduces a new penalty blending lasso and group lasso and develops an efficient algorithm for convex optimization without requiring orthonormal matrices.

Findings

The proposed algorithm efficiently solves the generalized sparse group lasso problem.

The method handles non-orthonormal model matrices effectively.

Solutions are sparse at both group and individual feature levels.

Abstract

We consider the group lasso penalty for the linear model. We note that the standard algorithm for solving the problem assumes that the model matrices in each group are orthonormal. Here we consider a more general penalty that blends the lasso (L1) with the group lasso ("two-norm"). This penalty yields solutions that are sparse at both the group and individual feature levels. We derive an efficient algorithm for the resulting convex problem based on coordinate descent. This algorithm can also be used to solve the general form of the group lasso, with non-orthonormal model matrices.