## Sharp Oracle Inequalities for Square Root Regularization

*Benjamin Stucky, Sara van de Geer*; 18(67):1−29, 2017.

### Abstract

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. (2010), are special cases of weakly
decomposable norms, therefore we also include the square root
LASSO (Belloni et al., 2011), the group square root LASSO (Bunea
et al., 2014) and a new method called the square root SLOPE (in
a similar fashion to the SLOPE from Bogdan et al. 2015). 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.

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