Towards the study of least squares estimators with convex penalty

Penalized least squares estimation is a popular technique in high-dimensional statistics. It includes such methods as the LASSO, the group LASSO, and the nuclear norm penalized least squares. The existing theory of these methods is not fully satisfying since it allows one to prove oracle inequalities with fixed high probability only for the estimators depending on this probability. Furthermore, the control of compatibility factors appearing in the oracle bounds is often not explicit. Some very recent developments suggest that the theory of oracle inequalities can be revised in an improved way. In this paper, we provide an overview of ideas and tools leading to such an improved theory. We show that, along with overcoming the disadvantages mentioned above, the methodology extends to the hilbertian framework and it applies to a large class of convex penalties. This paper is partly expository. In particular, we provide adapted proofs of some results from other recent work.