Group Lasso and Structured Sparsity

Think of a regression with one predictor for each of several industry-specific dummy indicators encoding a single categorical variable, or several lagged values of the same underlying series. Ordinary lasso treats each coefficient independently and can zero out some members of a natural group while keeping others — say, dropping "energy quarter-2" but keeping "energy quarter-3" — which produces a model that's hard to interpret and doesn't respect the fact that those coefficients only make sense together.

Group lasso fixes this by penalizing the size of an entire predefined group of coefficients as a single unit, rather than penalizing each coefficient on its own. Concretely, instead of summing the absolute value of each coefficient separately, the penalty sums the Euclidean norm of each group of coefficients — a construction that, like ordinary lasso, can push a whole group's norm to exactly zero once the penalty is strong enough, but does so for the entire group simultaneously rather than one coefficient at a time. So either all the industry dummies for a categorical variable survive, or none of them do.

This is the standard tool whenever predictors have a known group structure a researcher wants respected — categorical variables encoded as multiple dummies, polynomial terms, or a set of lagged values of the same factor — turning "which individual coefficients matter" into the more natural question "which whole variables matter."