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I have two questions about regularization:

  1. In the presence of high collinearity, ridge is better than Lasso, but if you need predictor selection, ridge is not what you want. In those cases, should you still use Lasso or is there any alternative (e.g., subset selection)?

  2. Can you use regularization on every predictive methods (regression, classification)?

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    $\begingroup$ Those are largely 2 different questions, which would probably go w/ 2 different threads. But I thought #2 was smaller & could serve as an add on, so I switched the order to make #1 your main question. Then I edited the title to highlight that. See if you think this is acceptable. $\endgroup$ – gung Mar 10 '17 at 3:02
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    $\begingroup$ You don't have to choose between the benefits of ridge and lasso. Look up the elastic net penalty. $\endgroup$ – gammer Mar 10 '17 at 3:04
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In the presence of high collinearity, ridge is better than Lasso, but if you need predictor selection, ridge is not what you want. In those cases, should you still use Lasso or is there any alternative (e.g., subset selection)?

Yes, there is an alternative that combines ridge and LASSO together called Elastic net. This minimizes the loss function:

$$ L = \sum_i (y_i - \hat y_i)^2 + \lambda \left( \alpha \sum_j \| \beta_j \| + (1 - \alpha) \frac{1}{2} \sum_j \beta_j^2 \right) $$

Here, $\lambda$ controls the overall regularization strength, and $\alpha$ is a number between zero and one (inclusive) that adjusts the relative strengths of the ridge vs. LASSO penalization.

I do not know of a situation where best subset selection is appropriate which does not arise from software or computing environment constraints.

Can you use regularization on every predictive methods (regression, classification)?

Yes. In the context of linear models, the elastic net is generalized by glmnet, which applies to any generalized linear model structure. For example, in logistic regression, the loss function for glmnet would be:

$$ L = \sum_i y_i \log(p_i) + (1 - y_i) \log(1 - p_i) + \lambda \left( \alpha \sum_j \| \beta_j \| + (1 - \alpha) \frac{1}{2} \sum_j \beta_j^2 \right) $$

These models are available in the glmnet package in R. You can learn to use the package here. You can read about how it works here.

There are other options as well, multilevel models can be seen as another way to apply regularization. That is well covered in the book by Gelman and Hill.

There are also a multitude of Bayesian approaches, here the choice of prior can be thought of as a regularization strategy.

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