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We study a family of sparse estimators defined as minimizers of someempirical Lipschitz loss function -- which include the hinge loss, the logisticloss and the quantile regression loss -- with a convex, sparse or group-sparseregularization. In particular, we consider the L1 norm on the coefficients, itssorted Slope version, and the Group L1-L2 extension. We propose a newtheoretical framework that uses common assumptions in the literature tosimultaneously derive new high-dimensional L2 estimation upper bounds for allthree regularization schemes. %, and to improve over existing results. For L1and Slope regularizations, our bounds scale as $(k^*/n) \log(p/k^*)$ --$n\times p$ is the size of the design matrix and $k^*$ the dimension of thetheoretical loss minimizer $\B{\beta}^*$ -- and match the optimal minimax rateachieved for the least-squares case. For Group L1-L2 regularization, our boundsscale as $(s^*/n) \log\left( G / s^* \right) + m^* / n$ -- $G$ is the totalnumber of groups and $m^*$ the number of coefficients in the $s^*$ groups whichcontain $\B{\beta}^*$ -- and improve over the least-squares case. We show that,when the signal is strongly group-sparse, Group L1-L2 is superior to L1 andSlope. In addition, we adapt our approach to the sub-Gaussian linear regressionframework and reach the optimal minimax rate for Lasso, and an improved ratefor Group-Lasso. Finally, we release an accelerated proximal algorithm thatcomputes the nine main convex estimators of interest when the number ofvariables is of the order of $100,000s$.

Improved error rates for sparse (group) learning with Lipschitz loss functions