A neutral comparison of algorithms to minimize L₀ penalties for high-dimensional variable selection

Variable selection methods based on L-0 penalties have excellent theoretical properties to select sparse models in a high-dimensional setting. There exist modifications of the Bayesian Information Criterion (BIC) which either control the familywise error rate (mBIC) or the false discovery rate (mBIC2) in terms of which regressors are selected to enter a model. However, the minimization of L-0 penalties comprises a mixed-integer problem which is known to be NP-hard and therefore becomes computationally challenging with increasing numbers of regressor variables. This is one reason why alternatives like the LASSO have become so popular, which involve convex optimization problems that are easier to solve. The last few years have seen some real progress in developing new algorithms to minimize L-0 penalties. The aim of this article is to compare the performance of these algorithms in terms of minimizing L-0-based selection criteria. Simulation studies covering a wide range of scenarios that are inspired by genetic association studies are used to compare the values of selection criteria obtained with different algorithms. In addition, some statistical characteristics of the selected models and the runtime of algorithms are compared. Finally, the performance of the algorithms is illustrated in a real data example concerned with expression quantitative trait loci (eQTL) mapping.