Fast Projections onto ℓ1, q -Norm Balls for Grouped Feature Selection

Joint sparsity is widely acknowledged as a powerful structural cue for performing feature selection in setups where variables are expected to demonstrate "grouped" behavior. Such grouped behavior is commonly modeled by Group-Lasso or Multitask Lasso-type problems, where feature selection is effected via l1,q -mixed-norms. Several particular formulations for modeling groupwise sparsity have received substantial attention in the literature; and in some cases, efficient algorithms are also available. Surprisingly, for constrained formulations of fundamental importance (e.g., regression with an l1,∞-norm constraint), highly scalable methods seem to be missing. We address this deficiency by presenting a method based on spectral projected-gradient (SPG) that can tackle l1,q -constrained convex regression problems. The most crucial component of our method is an algorithm for projecting onto l1,q>-norm balls. We present several numerical results which show that our methods attain up to 30X speedups on large l1,∞-multitask lasso problems. Even more dramatic are the gains for just the l1,∞-projection subproblem: we observe almost three orders of magnitude speedups compared against the currently standard method.