I am trying to adjust the penalty $\lambda$ in group lasso regression, but I have no idea about it. Just to clarify, group lasso regression is a kind of multiple linear regression which use penalties on estimated coefficients to keep them small. Also, it tries to assign same coefficients to variables which are in the same group.
Is there any theory or rule about maximum and minimum value of $\lambda$ based on input and response? I think the rule of $\lambda$ in lasso works for group lasso as well, so it is helpful.
I need an automatic procedure to determine the minimum and maximum value of penalty because I have more than 10 thousand response variables which regress on more than 500 independent variables.