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Recently, I started working on Ridge and Lasso regularization for Linear and Logistic Regression. My doubts are given below:

  1. Is the penalty the same (by same proportion) for all the coefficients or is it based on variable importance? If it is the latter I believe we can directly apply regularization rather than spending time in feature selection.
  2. Whether the Multi-collinearity is taken care by ridge and lasso regularization?

Thank you.

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    $\begingroup$ 1. The penalty is the $L_p$ norm, so large features are punished. The response variable is not taken into account. 2. LASSO is sensitive to it, so it is common to combine it with ridge regression to yield "elastic net" regression (L1+L2). This selects the entire collinear group. If you don't want that, drop the redundant ones first. $\endgroup$
    – Emre
    May 8, 2018 at 4:54
  • $\begingroup$ I think Emre answers the question. $\endgroup$
    – SmallChess
    May 8, 2018 at 5:02

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The penalty of both Lasso and Ridge is proportional to the magnitude of the weight. That is, the penalization added to the cost function is $\lambda ||\omega||_2$ or $\lambda ||\omega||_1$.

Wether it is more convenient to apply regularization or feature selection, Lasso already does some feature selection for you, as the estimated weights for Lasso are sparse (there will be many coefficients equal to 0).

About multi-colinearity, Ridge tends to eliminate variables that are colinear, while Lasso doesn't.

All that I have said is taken from An Introduction to Statistical Learning book, from James, Witten, Hastie and Tibshirani.

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  • $\begingroup$ Thanks for the reply. so if the magnitude is more then the penalty would be less and vice versa correct ? and also we need to perform normalization of the features when using Lasso and Ridge right ? $\endgroup$
    – deepguy
    May 8, 2018 at 15:08
  • $\begingroup$ Did I answer your question? $\endgroup$
    – David Masip
    May 8, 2018 at 15:09
  • $\begingroup$ Masp Yes you did. Thank you. Can you also please clarify the doubts i put in the previous question. $\endgroup$
    – deepguy
    May 8, 2018 at 15:13
  • $\begingroup$ Well, I saw it and it looks interesting, but I don't have a very good answer. All that I know is that in tree-based models colinearity is not a big problem. For this reason, I wouldn't worry. However, I do not know how to explain why. $\endgroup$
    – David Masip
    May 8, 2018 at 15:21

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