  We prove an L2 recovery bound for a family of sparse estimators defined as
minimizers of some empirical loss functions -- which include hinge loss and
logistic loss. More precisely, we achieve an upper-bound for coefficients
estimation scaling as (k*/n)\log(p/k*): n,p is the size of the design matrix
and k* the dimension of the theoretical loss minimizer. This is done under
standard assumptions, for which we derive stronger versions of a cone condition
and a restricted strong convexity. Our bound holds with high probability and in
expectation and applies to an L1-regularized estimator and to a recently
introduced Slope estimator, which we generalize for classification problems.
Slope presents the advantage of adapting to unknown sparsity. Thus, we propose
a tractable proximal algorithm to compute it and assess its empirical
performance. Our results match the best existing bounds for classification and
regression problems.
