Consider a model with a single predictor X1, no intercept term and an outcome y. We fit a lasso regression and obtain a coefficient for the predictor X1.

Now we create a new predictor X2 = X1 (you can imagine copying the column X1) and train another lasso model using the same regularization parameter as before. We will call the coefficients of the new model 1 and 2 (remember, there is no intercept term).

Compare the two models. It may be useful to consider the following points:

(1) can the models make the same predictions?;

(2) are there unique optimal coefficients 1,2?;

(3) are 1,2 related to ?