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When trying to predict data using linear regression or classify with logistic regression, with a polynomial, I know how to find the best degree of a polynomial to fits given data when the regularization coefficient is fixed. I also know how to find the best regularization coefficient when the degree of the polynomial is fixed.

What I want to know is how to find the best model when none of these parameters are known.

  • Should I find the best degree without regularization first, then the regularization parameter ?
  • Should I, for every degree, train with every possible regularization parameter value (assuming it belongs to an ensemble of discrete values), and then pick the combination degree/regularization that had the best results on the validation set ?

Or is there a better solution to find these hyperparameters ?

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    $\begingroup$ Is the context linear models with polynomial basis functions? $\endgroup$ – Matthew Drury Jan 29 '16 at 23:01
  • $\begingroup$ Yes, i updated the question to mention that $\endgroup$ – Carl Levasseur Jan 29 '16 at 23:49
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Since u are using regularization for feature selection I guess I would find the best regularization parameter for each order and then select the model with the smallest testing error. However, I think if u use a high order polynomial first and choose the best regularization parameter for that specific order you can get some insight.

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