# Group LASSO

Group LASSO selects explicitly defined groups of coefficients together: 

$$

  R(\beta) = \lambda\sum_{g=1}^K w_g\|\beta_g\|_2.

$$

Here $K$ is the number of groups. Groups must be disjoint, nonempty, and contain valid coefficient indices. `DisjointGroups` checks these requirements at construction. Supply one finite, nonnegative weight per group. Shrinkage applies those weights directly, without implicit group-size scaling. For a size adjustment, supply weights such as $w_g = \sqrt{|g|}$ explicitly. 

<a id="constraint-region"></a>

## Constraint region

![A disk for one two-coordinate group and a diamond for two singleton groups.](<../assets/ddb1fc99447ff4164738702519782fc8d7d0d9a5c3aeb9eca260adb871ed39de.svg>) 

Both panels use $\lambda = 1$ and unit weights. One group $\{1,2\}$ gives the unsquared L2 constraint $\sqrt{\beta_1^2 + \beta_2^2} \leq 1$. Two singleton groups give $|\beta_1| + |\beta_2| \leq 1$. The circle coincides with a ridge sublevel set in this example, although the penalty values and shrinkage differ. See the [plot conventions](<./#read-the-constraint-plots>). 

<a id="use-the-penalty"></a>

## Use the penalty

```rust
use shrinkage::{DisjointGroups, GroupLasso, Penalty, ProximalPenalty, UncoveredPolicy};

fn main() -> Result<(), Box<dyn std::error::Error>> {
    let groups = DisjointGroups::new(3, vec![vec![0, 2]], vec![1.0], UncoveredPolicy::Unpenalized)?;
    let penalty = GroupLasso::new(0.5, groups);
    assert_eq!(penalty.value(&[3.0, 9.0, 4.0])?, 2.5);
    let mut shrunk = [0.0; 3];
    penalty.prox(&[3.0, 9.0, 4.0], 1.0, &mut shrunk)?;
    assert_eq!(shrunk[1], 9.0);
    Ok(())
}
```

`UncoveredPolicy::Reject` requires complete coverage. `UncoveredPolicy::Unpenalized` leaves uncovered coefficients free. Zero-weight groups are also unpenalized. A group can become exactly zero because the proximal map shrinks its input vector by $\max(1 - s\lambda w_g/\|v_g\|_2,0)$, with zero input mapped to zero. 

Use [proximal gradient](<../reference/solvers/proximal-gradient>) with Gaussian or binomial datafits. The [multivariate model](<../models/multivariate>) shows how an axis penalty can select features across responses. 

API: [`shrinkage::GroupLasso`](<https://shrinkage.cc/packages/rust/reference/0e1357425403f686e1e3ace6b0f0af21e368cafd2bcaef0932eafd1ce2a4cf9c>) and [`shrinkage::DisjointGroups`](<https://shrinkage.cc/packages/rust/reference/66837f98e8b085ca419007cb1086d289a594d7573fda3cdd4d56b38563de68bc>).