SLOPE is Adaptive to Unknown Sparsity and Asymptotically Minimax

We consider high-dimensional sparse regression problems in which we observe $y = X \beta + z$, where $X$ is an $n \times p$ design matrix and $z$ is an $n$-dimensional vector of independent Gaussian errors, each with variance $\sigma^2$. Our focus is on the recently introduced SLOPE estimator ((Bogdan et al., 2014)), which regularizes the least-squares estimates with the rank-dependent penalty $\sum{1 \le i \le p} \lambdai |\hat \beta|{(i)}$, where $|\hat \beta|{(i)}$ 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 $\lambdai$ just about equal to $\sigma \cdot \Phi^{{-1}(1-iq/(2p))$} ($\Phi^{{-1}(\alpha)$} is the $\alpha$th quantile of a standard normal and $q$ is a fixed number in $(0,1)$) achieves a squared error of estimation obeying [ \sup{| \beta|0 \le k} \,\, \mathbb{P} \left(| \hat{\beta}{\text{SLOPE}} - \beta |² > (1+\epsilon) \, 2\sigma² k \log(p/k) \right) \longrightarrow 0 ] as the dimension $p$ increases to $\infty$, and where $\epsilon > 0$ is an arbitrary small constant. This holds under a weak assumption on the $\ell0$-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 $\ell0$-sparsity classes. We are not aware of any other estimator with this property.