  We leverage recent advances in high-dimensional statistics to derive new L2
estimation upper bounds for Lasso and Group Lasso in high-dimensions. For
Lasso, 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 ground truth
$\boldsymbol{\beta}^*$---and match the optimal minimax rate. For Group Lasso,
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 $\boldsymbol{\beta}^*$---and improve over existing results. We
additionally show that when the signal is strongly group-sparse, Group Lasso is
superior to Lasso.
