A number of regularization measures are available in literatures, which is kind of confusing to beginners. The classical penalty is ridge by Hoerl & Kennard (1970,Technometrics 12, 55–67).

enter image description here

Another modification to this is lasso by Tibshirani (1996, Journal of the Royal Statis- tical Society B 58, 267–288), defined as:

enter image description here

Another penalty is the elastic net penalty (Zou and Hastie 2005, Journal of the Royal Statistical Society B 67, 301–320) , which is a linear combination of the lasso penalty and the ridge penalty. Therefore the penalty covers these both as extreme cases.
enter image description here

The another penalty that I could find is bridge penalty introduced in Frank & Friedman (1993, Technometrics 35, 109–148). where λ ̃ = (λ, γ). It features an additional tuning parameter γ that controls the degree of preference for the estimated coefficient vector to align with the original, hence standardized, data axis direc- tions in the regressor space. It comprises the lasso penalty (γ = 1) and the ridge penalty (γ = 2) as special cases.

enter image description here

My question is : are there any preferences on type of penalty to use - something from or out of statistical text books ? Or this is just trial and error ? Please explain to layman language.

  • $\begingroup$ The no free lunch theorem might apply here? At least in terms of predictive power. The lasso penalty has the benefit of inducing sparseness if you're into that. $\endgroup$
    – user44764
    Commented Jul 21, 2014 at 19:59

1 Answer 1


There can be many considerations to this matter. To name a few:

  1. Inference: the distribution of ridge estimates is fairly simple to derive. Lasso, and basically any other penalty that performs variable selection, has only limited probabilistic results.
  2. Sparsity: If you desire a model with only a few predictors (say, for speed of prediction, for interpretability, ...) then you will want $l_1$ regularization.
  3. Speed of computation: The time complexity of the learning can be a consideration. There are differences between the algorithms. See here for some guidance. This becomes especially important if you plug the whole procedure in a cross validation scheme where models are fitted repeatedly.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.