Slope meets Lasso: improved oracle bounds and optimality

TL;DR

This paper demonstrates that Lasso and Slope estimators can adaptively achieve minimax prediction and estimation rates in high-dimensional linear regression under certain conditions, providing sharp oracle inequalities and optimal bounds.

Contribution

It establishes the minimax optimality of Lasso and Slope estimators with adaptive tuning under the RE condition, including non-asymptotic bounds and a comparative analysis of design assumptions.

Findings

Achieves minimax prediction and estimation rates $(s/n) \, \log(p/s)$.

Provides sharp oracle inequalities accounting for model misspecification.

Shows equivalence of RE and sparse eigenvalue conditions under bounded regressors.

Abstract

We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the exact minimax prediction and $\ell_2$ estimation rate $(s/n)\log (p/s)$ in high-dimensional linear regression on the class of $s$-sparse target vectors in $\mathbb R^p$. This is done under the Restricted Eigenvalue (RE) condition for the Lasso and under a slightly more constraining assumption on the design for the Slope. The main results have the form of sharp oracle inequalities accounting for the model misspecification error. The minimax optimal bounds are also obtained for the $\ell_q$ estimation errors with $1\le q\le 2$ when the model is well-specified. The results are non-asymptotic, and hold both in probability and in expectation. The assumptions that we impose on the design are satisfied with high probability for a large class of random matrices with independent and possibly anisotropically distributed rows. We give a comparative analysis of conditions, under which oracle bounds for the Lasso and Slope estimators can be obtained. In particular, we show that several known conditions, such as the RE condition and the sparse eigenvalue condition are equivalent if the $\ell_2$-norms of regressors are uniformly bounded.