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In exciting new work, Bertsimas et al. (2016) showed that the classical bestsubset selection problem in regression modeling can be formulated as a mixedinteger optimization (MIO) problem. Using recent advances in MIO algorithms,they demonstrated that best subset selection can now be solved at much largerproblem sizes that what was thought possible in the statistics community. Theypresented empirical comparisons of best subset selection with other popularvariable selection procedures, in particular, the lasso and forward stepwiseselection. Surprisingly (to us), their simulations suggested that best subsetselection consistently outperformed both methods in terms of predictionaccuracy. Here we present an expanded set of simulations to shed more light onthese comparisons.  The summary is roughly as follows: (a) neither best subset selection nor thelasso uniformly dominate the other, with best subset selection generallyperforming better in high signal-to-noise (SNR) ratio regimes, and the lassobetter in low SNR regimes; (b) best subset selection and forward stepwiseperform quite similarly throughout; (c) the relaxed lasso (actually, asimplified version of the original relaxed estimator defined in Meinshausen,2007) is the overall winner, performing just about as well as the lasso in lowSNR scenarios, and as well as best subset selection in high SNR scenarios.

Extended Comparisons of Best Subset Selection, Forward Stepwise Selection, and the Lasso