SLOPE—Adaptive variable selection via convex optimization

We introduce a new estimator for the vector of coefficients in the linear model y=X+z, where X has dimensions n p with p possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to \₁^₏12 y-Xb_₂^2+₁ b (₁) +₂ b (₂) ++₏ b (₏), \ where ₁₂₏0 and b (₁) b (₂) b (₏) are the decreasing absolute values of the entries of b. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ₁ procedures such as the Lasso. Here, the regularizer is a sorted ₁ norm, 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. B 57 (1995) 289–300 procedure (BH) which compares more significant p-values with more stringent thresholds. One notable choice of the sequence \₈\ is given by the BH critical values ₁₇ (i) =z (1-i q/2p), where q (0, 1) and z () 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 ₁₇ provably controls FDR at level q. Moreover, it also appears to have appreciable inferential properties under more general designs X while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.