An error bound for Lasso and Group Lasso in high dimensions

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× p is the size of the design matrix and k^* the dimension of the ground truth boldsymbolβ^*---and match the optimal minimax rate. For Group Lasso, our bounds scale as (s^*/n) logleft( G / s^* right) + m^* / n---G is the total number of groups and m^* the number of coefficients in the s^* groups which contain boldsymbolβ^*---and improve over existing results. We additionally show that when the signal is strongly group-sparse, Group Lasso is superior to Lasso.