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There has been an explosion of interest in using $l_1$-regularization inplace of $l_0$-regularization for feature selection. We present theoreticalresults showing that while $l_1$-penalized linear regression never outperforms$l_0$-regularization by more than a constant factor, in some cases using an$l_1$ penalty is infinitely worse than using an $l_0$ penalty. We also showthat the "optimal" $l_1$ solutions are often inferior to $l_0$ solutions foundusing stepwise regression.  We also compare algorithms for solving these two problems and show thatalthough solutions can be found efficiently for the $l_1$ problem, the"optimal" $l_1$ solutions are often inferior to $l_0$ solutions found usinggreedy classic stepwise regression. Furthermore, we show that solutionsobtained by solving the convex $l_1$ problem can be improved by selecting thebest of the $l_1$ models (for different regularization penalties) by using an$l_0$ criterion. In other words, an approximate solution to the right problemcan be better than the exact solution to the wrong problem.

A Risk Ratio Comparison of $l_0$ and $l_1$ Penalized Regression