SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

We consider high-dimensional sparse regression problems in which we observe $y = X β+ z$, where $X$ is an $n \times p$ design matrix and $z$ is an $n$-dimensional vector of independent Gaussian errors, each with variance $σ^2$. Our focus is on the recently introduced SLOPE estimator ((Bogdan et al., 2014)), which regularizes the least-squares estimates with the rank-dependent penalty $\sum_{1 \le i \le p} λ_i |\hat β|_{(i)}$, where $|\hat β|_{(i)}$ is the $i$th largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of $X$ are i.i.d.~$\mathcal{N}(0, 1/n)$, we show that SLOPE, with weights $λ_i$ just about equal to $σ\cdot Φ^{-1}(1-iq/(2p))$ ($Φ^{-1}(α)$ is the $α$th quantile of a standard normal and $q$ is a fixed number in $(0,1)$) achieves a squared error of estimation obeying \[ \sup_{\| β\|_0 \le k} \,\, \mathbb{P} \left(\| \hatβ_{\text{SLOPE}} - β\|^2 > (1+ε) \, 2σ^2 k \log(p/k) \right) \longrightarrow 0 \] as the dimension $p$ increases to $\infty$, and where $ε> 0$ is an arbitrary small constant. This holds under a weak assumption on the $\ell_0$-sparsity level, namely, $k/p \rightarrow 0$ and $(k\log p)/n \rightarrow 0$, 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 $\ell_0$-sparsity classes. We are not aware of any other estimator with this property.