Abstract

We consider the linear model and the Lasso estimator. Our goal is to provide upper and lower bounds for the prediction error that are close to each other. We assume that the active components of the vector of regression coefficients are sufficiently large in absolute value (in a sense that will be specified) and that the tuning parameter is suitably chosen. The bounds depend on so-called compatibility constants. We will present the definition of the compatibility constants and discuss their relation with restricted eigenvalues. As an example, we consider the the least squares estimator with total variation penalty and present bounds with small gap. For the case of random design, we assume that the rows of the design matrix are i.i.d.copies of a Gaussian random vector. We assume that the largest eigenvalue of the covariance matrix remains bounded and establish under some mild compatibility conditions upper and lower bounds with ratio tending to one.