The LASSO problem works by minimizing
$$\min_\beta (\frac{1}{2}\left\rVert y-X\beta\right\rVert^2_2+\lambda\left\rVert\beta\right\rVert_1)$$
Here in this webpage I found that the minimal value of the $\lambda$ parameter that makes $0$ all the $\beta$ coefficients in the model is computed as $\lambda=\left\lVert X^ty\right\rVert_\infty$, so any value of $\lambda$ greater than this will send all the $\beta$ coefficients to zero.
On the other side, if our variables are grouped into $m$ groups of size $p_l$ the group LASSO works by solving:
$$\min_{\vec{\beta}}\left\{\frac{1}{2} \left\lVert\vec{y}-\sum_{l=1}^mX^{(l)}\vec{\beta^{(l)}}\right\rVert_2^2 +\lambda\sum_{l=1}^m\sqrt{p_l}\left\lVert\vec{\beta^{(l)}}\right\rVert_2\right\},$$
Is there a way to find the minimal value for the $\lambda$ parameter in the case of the group LASSO? Consider for instance the gglasso package and the BostonHousing dataset. I will consider a random grouping structure for the data:
library(mlbench, gglasso)
data("BostonHousing")
x <- data.matrix(BostonHousing[,-14])
y <- BostonHousing[,14]
index <- c(1,1,2,2,2,2,2,2,2,3,3,3,3)
fit = gglasso(x, y, loss="ls", nlambda=5, intercept=TRUE, group = index)
fit$lambda[1]
> 389.1944
fit$beta[,1]
> crim zn indus chas nox rm age dis rad tax ptratio b lstat
0 0 0 0 0 0 0 0 0 0 0 0 0
I would really appreciate any insight, article or book that shows how to find this minimal lambda, and that demonstrates why is it computed that way, not just for the group LASSO but also for the LASSO.