I want to run a penalized multinomial logit and logit regression using the glmnet package in R. I understand, that before fitting the penalized model, one should standardize the variables, to penalize each coefficient equally.
There are posts which address similar topics but not how the mathematical process is designed:
https://think-lab.github.io/d/205/#5
Skimming through the glmnet vignette, I found that the variables are standardized per default. What I don't understand is the phase: "The coefficients are always returned on the original scale" - how is this done mathematically?
So what I understand is:
standardize all variables $x_i$ : $\hat{x_i}=\frac{x_i- \bar{x_i}}{\sigma_{x_i}}$
obtain $\hat{\beta}= \arg\max \log(L) - \lambda \cdot 0.5 \cdot \Vert\beta\Vert_2^2$
So my question is:
How is the outlined process to be complemented, in order to arrive at the final reported estimates?
