Bounds on the prediction error of penalized least squares estimators with convex penalty

Understand

This paper considers the penalized least squares estimator with arbitrary convex penalty.

• When the observation noise is Gaussian, we show that the prediction error is a subgaussian random variable concentrated around its median.
• We apply this concentration property to derive sharp oracle inequalities for the prediction error of the LASSO, the group LASSO and the SLOPE estimators, both in probability and in expectation.
• In contrast to the previous work on the LASSO type methods, our oracle inequalities in probability are obtained at any confidence level for estimators with tuning parameters that do not depend on the confidence level.