  We study a family of sparse estimators defined as minimizers of some
empirical Lipschitz loss function -- which include the hinge loss, the logistic
loss and the quantile regression loss -- with a convex, sparse or group-sparse
regularization. In particular, we consider the L1 norm on the coefficients, its
sorted Slope version, and the Group L1-L2 extension. We propose a new
theoretical framework that uses common assumptions in the literature to
simultaneously derive new high-dimensional L2 estimation upper bounds for all
three regularization schemes. %, and to improve over existing results. For L1
and Slope regularizations, our bounds scale as $(k^*/n) \log(p/k^*)$ --
$n\times p$ is the size of the design matrix and $k^*$ the dimension of the
theoretical loss minimizer $\B{\beta}^*$ -- and match the optimal minimax rate
achieved for the least-squares case. For Group L1-L2 regularization, our bounds
scale as $(s^*/n) \log\left( G / s^* \right) + m^* / n$ -- $G$ is the total
number of groups and $m^*$ the number of coefficients in the $s^*$ groups which
contain $\B{\beta}^*$ -- and improve over the least-squares case. We show that,
when the signal is strongly group-sparse, Group L1-L2 is superior to L1 and
Slope. In addition, we adapt our approach to the sub-Gaussian linear regression
framework and reach the optimal minimax rate for Lasso, and an improved rate
for Group-Lasso. Finally, we release an accelerated proximal algorithm that
computes the nine main convex estimators of interest when the number of
variables is of the order of $100,000s$.
