Error bounds for the convex loss Lasso in linear models
Abstract In this paper we investigate error bounds for convex loss functions for the Lasso in linear models, by first establishing a gap in the theory with respect to the existing error bounds. Then, under the compatibility condition, we recover bounds for the absolute value estimation error and the squared prediction error under mild conditions, which appear to be far more appropriate than the existing bounds for the convex loss Lasso. Interestingly, asymptotically the only difference between the new bounds of the convex loss Lasso and the classical Lasso is a term solely depending on a well-known expression in the robust statistics literature appearing multiplicatively in the bounds. We show that this result holds whether or not the scale parameter needs to be estimated jointly with the regression coefficients. Finally, we use the ratio to optimize our bounds in terms of minimaxity.
Keywords Robust Lasso, high dimensions, error bounds,
joint scale and location estimation
Citation Hannay, M., & Deléamont, P. Y. (2017). Error bounds for the convex loss Lasso in linear models. Electronic Journal of Statistics, 11(2), 2832-2875.
Type Journal article (English)
Date of appearance 2017
Journal Electronic Journal of Statistics
Volume 11
Issue 2
Pages 2832-2875