Solved – Regularized Logistic Regression: Lasso vs. Ridge vs. Elastic Net

Since I'm relatively new to regularized regressions, I'm concerned with the hughe differences lasso, ridge and elastic nets deliver.

My data set has the following characteristics:

  • panel data set: > 900.000 obs. and over 50 variables
  • highly unbalances
  • 2-5 variables are highly correlated.

To select only a subset of the variables I used penalized logistic regression fitting the model:
$frac{1}{N} sum_{i=1}^{N}L(beta,X,y)-lambda[(1-alpha)||beta||^2_2/2+alpha||beta||_1] $

To determine the optimal $lambda$ I used cross validation which yileds the following results:

enter image description here
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The elastic net looks quite similar to the Lasso, also proposing only 2 Variables.

So my main question is: why do these approaches deliver so different results?
According to the Lasso, I only do have 2 variables in the final model and according to the Ridge, I do have 34 variables?

So in the end – which approach is the right one?
And why are the results so extremely different?

Thanks a lot!

By mean squared error do you mean the Brier score? And for elastic net the plot should be 3-dimensional since there are 2 simultaneous penalty parameters. Don't force $alpha$ to be 0 or 1.

To answer your question, the lasso is spending information trying to be parsimonious, while a quadratic penalty is not trying to select features but is just trying to predict accurately. It is a fools errand to expect that a typical problem will result in a parsimonious model that is highly discriminating. In addition, the lasso is not stable, i.e., if you were to repeat the experiment the list of selected features would vary quite a lot.

For optimum prediction use ridge logistic regression. Elastic net is a nice compromise between that and lasso.

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