If we want to conduct variable selection for a high-dimensional data with the binary responses, one good solution will be using L1 regularized logistic regression.
However, I wonder what will happen if I use L1 regularized linear regression (i.e. Lasso) for the binary response data. Particularly, I am hoping to find some statistical analysis of such a model misspecification. For example, given the oracle $\lambda$ (the regularization weight), what will the differences of estimating the effect sizes with these two methods.
I thought there are some papers discussing this, but I couldn't find any.