# Improved bounds for Square-Root Lasso and Square-Root Slope

## TL;DR

This paper extends the theoretical understanding of Square-Root Lasso and Square-Root Slope estimators, demonstrating their optimal prediction and estimation rates in high-dimensional sparse linear regression with unknown variance, and establishing their adaptivity and concentration properties.

## Contribution

The paper proves that Square-Root Lasso and Square-Root Slope achieve optimal rates in high-dimensional sparse regression with unknown variance, extending previous results and establishing adaptivity and non-asymptotic bounds.

## Key findings

- Achieve minimax prediction rate $(s/n) \\log (p/s)$ under mild conditions.
- Optimal estimation error bounds in $l_q$-norms for both estimators.
- Establish adaptivity to unknown variance and sparsity, with valid confidence rates.

## Abstract

Extending the results of Bellec, Lecu\'e and Tsybakov to the setting of sparse high-dimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is $(s/n) \log (p/s)$, up to some constant, under some mild conditions on the design matrix. Here, $n$ is the sample size, $p$ is the dimension and $s$ is the sparsity parameter. We also prove optimality for the estimation error in the $l_q$-norm, with $q \in [1,2]$ for the Square-Root Lasso, and in the $l_2$ and sorted $l_1$ norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity $s$ of the true parameter. Next, we prove that any estimator depending on $s$ which attains the minimax rate admits an adaptive to $s$ version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [Bellec, Lecu\'e and Tsybakov, 2017] where the case of known variance is treated. Our results are non-asymptotic.