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We study a seemingly unexpected and relatively less understood overfittingaspect of a fundamental tool in sparse linear modeling - best subset selection,which minimizes the residual sum of squares subject to a constraint on thenumber of nonzero coefficients. While the best subset selection procedure isoften perceived as the "gold standard" in sparse learning when the signal tonoise ratio (SNR) is high, its predictive performance deteriorates when the SNRis low. In particular, it is outperformed by continuous shrinkage methods, suchas ridge regression and the Lasso. We investigate the behavior of best subsetselection in the high-noise regimes and propose an alternative approach basedon a regularized version of the least-squares criterion. Our proposedestimators (a) mitigate, to a large extent, the poor predictive performance ofbest subset selection in the high-noise regimes; and (b) perform favorably,while generally delivering substantially sparser models, relative to the bestpredictive models available via ridge regression and the Lasso. We conduct anextensive theoretical analysis of the predictive properties of the proposedapproach and provide justification for its superior predictive performancerelative to best subset selection when the noise-level is high. Our estimatorscan be expressed as solutions to mixed integer second order conic optimizationproblems and, hence, are amenable to modern computational tools frommathematical optimization.

Subset Selection with Shrinkage: Sparse Linear Modeling when the SNR is low