Earlier I asked whether grid fineness of $\lambda$ is related to overfitting in LASSO, ridge regression and elastic net models. I got an answer that it is not the case. Now I am asking,
Question: Is grid fineness of $\alpha$ in elastic net related to overfitting?
($\alpha$ is the parameter governing the balance between $L_1$ and $L_2$ penalty.)
The argumentation in the answer to the linked question goes like this:
we definitely want to optimize our penalized likelihood function for values $\lambda$, and it doesn't matter how many values of $\lambda$ we test, because out-of-sample performance for a fixed data set and fixed partitioning is entirely deterministic. More to the point, the out-of-sample metric is not at all altered by how many values $\lambda$ you test.
I would guess that the same applies to $\alpha$ in place of $\lambda$, and hence a finer grid can only help but not hurt. Is that right?
(A note may be due that when doing cross validation, I fix $\alpha$ first and then do a search over a grid of $\lambda$s.)