  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.
