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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 allready some posts which adress similar topics - but they don't answer the question how the mathematical process is designed:

Coefficient value from glmnet

https://stackoverflow.com/questions/41122803/how-does-glmnet-standardize-variables-when-weights-are-present

https://think-lab.github.io/d/205/#5

Skimming through the glmnet vignette, I could find 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:

  1. standardize all variables $x_i$ : $\hat{x_i}=\frac{x_i- \bar{x_i}}{\sigma_{x_i}}$

  2. obtain $\hat{\beta}= \text{argmax } log(L) - \lambda \cdot 0.5 \cdot ||\beta||_2^2$

So my quiestion basically is:

How is the outlined process to be complemented, in order to arive at the final reported estimates?

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