A note on sharp oracle bounds for Slope and Lasso

TL;DR

This paper extends the theoretical understanding of Slope and Lasso estimators by deriving sharp oracle bounds under less restrictive conditions, accommodating nearly sparse parameters and providing optimal error bounds across various norms.

Contribution

It generalizes existing results to non-exactly sparse vectors and establishes optimal bounds for $\, ext{ell}_q$ errors using extended Restricted Eigenvalue conditions.

Findings

Derived sharp oracle bounds for Slope and Lasso.

Extended theoretical results to nearly sparse vectors.

Established optimal $\, ext{ell}_q$ error bounds.

Abstract

In this paper, we study the sharp oracle bounds for Slope and Lasso and generalize the results in Bellec et al. (2018) to allow the case that the parameter vector is not exactly sparse and obtain the optimal bounds for $\ell_q$ estimation errors with $1\leq q\leq \infty$ by using some extended Restricted Eigenvalue type conditions.