SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

We consider high-dimensional sparse regression problems in which we observe y = Xβ + z,where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussianerrors, each with variance σ. Our focus is on the recently introduced SLOPE estimator [15],which regularizes the least-squares estimates with the rank-dependent penalty∑1≤i≤p λi|β̂|(i),where|β̂|(i) is the ith largest magnitude of the fitted coefficients. Under Gaussian designs, wherethe entries of X are i.i.d. N (0, 1/n), we show that SLOPE, with weights λi just about equal toσ · Φ−1(1− iq/(2p)) (Φ−1(α) is the αth quantile of a standard normal and q is a fixed numberin (0, 1)) achieves a squared error of estimation obeying sup‖β‖0≤kP(‖β̂SLOPE − β‖ > (1 + ) 2σk log(p/k))−→ 0 as the dimension p increases to ∞, and where > 0 is an arbitrary small constant. This holdsunder weak assumptions on the sparsity level k and is sharp in the sense that this is the bestpossible error any estimator can achieve. A remarkable feature is that SLOPE does not requireany knowledge of the degree of sparsity, and yet automatically adapts to yield optimal totalsquared errors over a wide range of sparsity classes. We are not aware of any other estimatorwith this property.