title: SLOPE-ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION.
abstract: We introduce a new estimator for the vector of coefficientsβin the linear modely=Xβ+z, whereXhas dimensionsn × pwithppossibly larger thann. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to [Formula: see text]where λ1≥ λ2≥ … ≥ λp≥ 0 and [Formula: see text] are the decreasing absolute values of the entries ofb. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ1procedures such as the Lasso. Here, the regularizer is a sorted ℓ1norm, which penalizes the regression coefficients according to their rank: the higher the rank-that is, stronger the signal-the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B57(1995) 289-300] procedure (BH) which compares more significantp-values with more stringent thresholds. One notable choice of the sequence {λi} is given by the BH critical values [Formula: see text], whereq∈ (0, 1) andz(α) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λBHprovably controls FDR at levelq. Moreover, it also appears to have appreciable inferential properties under more general designsXwhile having substantial power, as demonstrated in a series of experiments running on both simulated and real data.
authors.affiliation: Department of Mathematics, Wrocław University of Technology, 50-370 Wrocław, Poland.
authors.forename: Małgorzata
authors.initials: M
authors.lastname: Bogdan
authors.affiliation: Human Language Technologies, IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA.
authors.forename: Ewout
authors.initials: E
authors.lastname: van den Berg
authors.affiliation: Department of Health Research and Policy, Division of Biostatistics, Stanford University, HRP Redwood Building, Stanford, California 94305, USA.
authors.forename: Chiara
authors.initials: C
authors.lastname: Sabatti
authors.affiliation: Department of Statistics, Stanford University, 90 Serra Mall, Sequoia Hall, Stanford, California 94305, USA.
authors.forename: Weijie
authors.initials: W
authors.lastname: Su
authors.affiliation: Department of Statistics, Stanford University, 390 Serra Mall, Sequoia Hall, Stanford, California 94305, USA.
authors.forename: Emmanuel J
authors.initials: EJ
authors.lastname: Candès
keywords: Lasso; Sparse regression; false discovery rate; sorted ℓ1 penalized estimation (SLOPE); variable selection