Sharp Oracle Inequalities for Square Root Regularization

We study a set of regularization methods for high-dimensional linear regression models. These penalized estimators have the square root of the residual sum of squared errors as loss function, and any weakly decomposable norm as penalty function. This fit measure is chosen because of its property that the estimator does not depend on the unknown standard deviation of the noise. On the other hand, a generalized weakly decomposable norm penalty is very useful in being able to deal with different underlying sparsity structures. We can choose a different sparsity inducing norm depending on how we want to interpret the unknown parameter vector $\beta$. Structured sparsity norms, as defined in Micchelli et al. [18], are special cases of weakly decomposable norms, therefore we also include the square root LASSO (Belloni et al. [3]), the group square root LASSO (Bunea et al. [10]) and a new method called the square root SLOPE (in a similar fashion to the SLOPE from Bogdan et al. [6]). For this collection of estimators our results provide sharp oracle inequalities with the Karush-Kuhn-Tucker conditions. We discuss some examples of estimators. Based on a simulation we illustrate some advantages of the square root SLOPE

Square Root {LASSO}: well-posedness, Lipschitz stability and the tuning trade off

This paper studies well-posedness and parameter sensitivity of the Square Root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over the standard LASSO) is that the optimal tuning of the regularization parameter is robust with respect to measurement noise. This paper provides three point-based regularity conditions at a solution of the SR-LASSO: the weak, intermediate, and strong assumptions. It is shown that the weak assumption implies uniqueness of the solution in question. The intermediate assumption yields a directionally differentiable and locally Lipschitz solution map (with explicit Lipschitz bounds), whereas the strong assumption gives continuous differentiability of said map around the point in question. Our analysis leads to new theoretical insights on the comparison between SR-LASSO and LASSO from the viewpoint of tuning parameter sensitivity: noise-robust optimal parameter choice for SR-LASSO comes at the "price" of elevated tuning parameter sensitivity. Numerical results support and showcase the theoretical findings