Introduction

Abstract

We start with a brief overview of empirical risk minimization problems and of the role of empirical and Rademacher processes in constructing distribution dependent and data dependent excess risk bounds. We then discuss penalized empirical risk minimization and oracle inequalities and conclude with sparse recovery and low rank matrix recovery problems. Many important aspects of empirical risk minimization are beyond the scope of these notes, in particular, the circle of questions related to approximation theory (see well known papers by Cucker and Smale [47], DeVore et al. [49] and references therein).