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

arXiv admin note: text overlap with arXiv:1910.08880