How to find uppper bound for the hyper parameters in linear model?

Question

I have a loss function of the following form:

$$\arg\min_w= \|Y-w^TX\|_2^2 + \lambda\|w\|_1 + \gamma \sum_{g=1}^G\|w_g\|_1$$

where $w_g$ is a group of coefficients. Given $Y$ and $X$, I would like to find upper bound for $\lambda$ and/or $\gamma$ that will set all weights ($w$) to zero. can anyone help me how to drive such upper bound for this loss function?

Answer 1

In the lasso-problem a parameter $\lambda \ge \|X^T Y\|_\infty$ will set all regression coefficients to zero. See, for example, page 813 of

Dossal, Charles, et al. "The degrees of freedom of the Lasso for general design matrix." Statistica Sinica (2013): 809-828.

as a reference for this.