  We obtain bounds on estimation error rates for regularization procedures of
the form \begin{equation*}
  \hat f \in {\rm argmin}_{f\in
  F}\left(\frac{1}{N}\sum_{i=1}^N\left(Y_i-f(X_i)\right)^2+\lambda
\Psi(f)\right) \end{equation*} when $\Psi$ is a norm and $F$ is convex.
  Our approach gives a common framework that may be used in the analysis of
learning problems and regularization problems alike. In particular, it sheds
some light on the role various notions of sparsity have in regularization and
on their connection with the size of subdifferentials of $\Psi$ in a
neighbourhood of the true minimizer.
  As `proof of concept' we extend the known estimates for the LASSO, SLOPE and
trace norm regularization.
