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I was building a OLS model. The RMSE was not very satisfactory. So I tried feature engineering and the result improved and there was no over-fitting. To, further improve the results I thought of trying Ridge regression, but there was no significant improvement.

So, my question is - Do regularization techniques work towards improving the predictability ONLY by reducing variance? If your regression model does not have high variance then using regularization will not help?

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Very good question. Let me break down the answer in two parts.

If your regression model does not have high variance then using regularization will not help?

As you have empirically found out, it won't necessarily help. A simple linear model with a healthy (i.e. large) number of samples per dimension is already simple enough, so regularisation is not guaranteed to improve results.

I guess the answer to the question also depends on what you are building your model for. In statistics you typically have a reason for fitting a particular model with a particular set of variables, and regularisation can be sometimes unnecessary and sometimes even undesirable. In machine learning, in contrast, the common wisdom is that you should:

  1. Make your model complex enough so that it overfits; and
  2. Then add regularisation so that it doesn't overfit.

So, if you're only looking for a better MSE (as I understand from the first line in your question) then follow the machine learning approach: make a bigger model and then regularise it.

Do regularization techniques work towards improving the predictability ONLY by reducing variance?

The short answer is no. Regularisation works in various ways, more mysterious the more complex your model is. This is because a regularisation term can interfere with your optimisation or model selection algorithms, so saying it acts "ONLY by reducing variance" is overly simplistic.

Overall, let me emphasise that regularisation is mostly an ad-hoc process, and therefore never guaranteed to work -- although it is always worth trying. There are cases (especially with bigger models) in which even if your model doesn't overfit you still get better performance with a carefully tuned regulariser.

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  • $\begingroup$ So, if I say that " it is good to use regularization when you do not want to go through the pain of carefully selecting variables but do not always bank on regularization to improve prediction " would this be a valid statement? $\endgroup$ – saurav shekhar May 25 '17 at 9:18
  • $\begingroup$ @sauravshekhar The first half of your statement ("it is good to use regularization when you do not want to go through the pain of carefully selecting variables") is perfectly fine. The second half ("do not always bank on regularization to improve prediction") is partially correct, in the sense that you shouldn't always expect regularisation to improve prediction but you should definitely always try it. $\endgroup$ – Pedro Mediano May 25 '17 at 11:01

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