An error bound for Lasso and Group Lasso in high dimensions

Abstract: 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 $k<sup>*$ the dimension of the ground truth $\boldsymbol{\beta}<sup>*$---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 $m<sup>*$ the number of coefficients in the $s<sup>*$ groups which contain $\boldsymbol{\beta}<sup>*$---and improve over existing results. We additionally show that when the signal is strongly group-sparse, Group Lasso is superior to Lasso.