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I have just learned that a general framework in constrained optimization is called "proximal gradient optimization". It is interesting that the $\ell_0$ "norm" is also associated with a proximal operator. Hence, one can apply iterative hard thresholding algorithm to get the sparse solution of the following

$$\min \Vert Y-X\beta\Vert_F + \lambda \vert \beta \vert_0$$

If so, why people are still using $\ell_1$? If you can just get the result by non-convex optimization directly, why are people still using LASSO?

I want to know what's the downside of the proximal gradient approach for $\ell_0$ minimization. Is it because of the non-convexity and randomness associated with? That means the initial estimator is very important.

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    $\begingroup$ What is the goal for which you consider lasso vs. proximal gradient approach to $l_0$ minimization as solutions? $\endgroup$ – Richard Hardy Oct 15 '19 at 18:25
  • $\begingroup$ @RichardHardy the goal is to understand what are the pros and cons. If one thing doesn't eliminate the other, it must be the case that both has advantages over certain problems. $\endgroup$ – ArtificiallyIntelligence Oct 16 '19 at 12:59
  • $\begingroup$ It is difficult to discuss pros and cons when there is no goal of applying the methods. If you do not have a concrete goal, how do you eveluate whether a method has performed well or poorly? $\endgroup$ – Richard Hardy Oct 16 '19 at 16:28
  • $\begingroup$ @RichardHardy say for instance, the computational efficency, convergence property... is it giving false positive over a set of exmaples... $\endgroup$ – ArtificiallyIntelligence Oct 16 '19 at 23:46
  • $\begingroup$ These are features of the methods, but what goal are they supposed to be targeting? Prediction under some specific loss? Variable selection under some specific loss (characterizin how costly inclusion of irrelevant regressors vs. exclusion of relevant ones are)? $\endgroup$ – Richard Hardy Oct 17 '19 at 5:15

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