# What's the rationale behind multiple response LASSO?

I understand that, with LASSO, the regularization term puts a constraint on the complexity of our regression model. Usually, for prediction applications, regularization makes the model perform better in out of sample data set.

What I cannot understand is that, why regularization term could help in a multi-response setup.

For instance, we have 2 predictors, $$f_{1}$$ and $$f_{2}$$, and we have 2 responses, $$y_{1}$$ and $$y_{2}$$, we are searching for a 2 x 2 matrix of betas, where $$\beta_{1}^{1}$$ and $$\beta_{1}^{2}$$ represent the coefficient of $$f_{1}$$ for the first and second responses.

Simply speaking, there are two scenarios, first: the responses are correlated, then we expect that the coefficients $$\beta_{1}^{1}$$ and $$\beta_{1}^{2}$$ should be similar. The second scenario, responses are not correlated or negatively correlated, $$\beta_{1}^{1}$$ and $$\beta_{1}^{2}$$ should be distinct.

What I understand after reading glmnet doct is that, through the regularization term, we enforce that the norm of vector $$[\beta_{1}^{1}, \beta_{1}^{2}]$$ is small. However, this does NOT enforce any relationship as we described above, right?

Therefore, my question is how could such 'joint' regularization constraint better than 2 individual LASSO fits? or in other words, what could go wrong if we run 2 LASSO fits for $$y_{1}$$ and $$y_{2}$$ individually?

Just to make sure we are all talking about the same thing: in the linked document, the penalty term is

$$(1-\alpha)||\beta||_{F}^{2}/2 + \alpha\sum_{j=1}^{p}||\beta_{j}||_{2}$$

using GLMNET's terminology , when $$\alpha = 1$$, we have LASSO

so the penalty term is $$\alpha\sum_{j=1}^{p}||\beta_{j}||_{2}$$, which is the collection of $$L2$$ normal of each row of betas, i.e., betas for each response for a specific predictor