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 Gaussian errors, each with variance σ2. Our focus is on the recently introduced SLOPE estimator [Ann. Appl. Stat. 9 (2015) 1103–1140], which regularizes the least-squares estimates with the rank-dependent penalty ∑1≤i≤pλi|βˆ|(i), where |βˆ|(i) is the ith largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of X are i.i.d. N(0,1/n), we show that SLOPE, with weights λi just about equal to σ⋅Φ−1(1−iq/(2p)) (Φ−1(α) is the αth quantile of a standard normal and q is a fixed number in (0,1)) achieves a squared error of estimation obeying sup‖β‖0≤kℙ(‖βˆSLOPE−β‖2>(1+ε)2σ2k log(p/k)) ⟶ 0 as the dimension p increases to ∞, and where ε>0 is an arbitrary small constant. This holds under a weak assumption on the sparsity level k and is sharp in the sense that this is the best possible error any estimator can achieve. A remarkable feature is that SLOPE does not require any knowledge of the degree of sparsity, and yet automatically adapts to yield optimal total squared errors over a wide range of sparsity classes. We are not aware of any other estimator with this property.