An error bound for Lasso and Group Lasso in high dimensions
This work provides theoretical guarantees for sparse regression methods, which is incremental for researchers in high-dimensional statistics and machine learning.
The paper derived new L2 estimation upper bounds for Lasso and Group Lasso in high-dimensional settings, showing that Lasso matches the optimal minimax rate with bounds scaling as $(k^*/n) \\log(p/k^*)$ and Group Lasso improves over existing results with bounds scaling as $(s^*/n) \\log(G/s^*) + m^*/n$, and demonstrated Group Lasso's superiority for strongly group-sparse signals.
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 $\boldsymbolβ^*$---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 $\boldsymbolβ^*$---and improve over existing results. We additionally show that when the signal is strongly group-sparse, Group Lasso is superior to Lasso.