A Theory of the Risk for Optimization with Relaxation and its Application to Support Vector Machines

In this paper we consider optimization with relaxation, an ample paradigm to make data-driven designs. This approach was previously considered by the same authors of this work in Garatti and Campi (2019), a study that revealed a deep-seated connection between two concepts: risk (probability of not satisfying a new, out-of-sample, constraint) and complexity (according to a definition introduced in paper Garatti and Campi (2019)). This connection was shown to have profound implications in applicat...