Scenario optimization with relaxation: a new tool for design and application to machine learning problems

Scenario optimization is by now a well established technique to perform designs in the presence of uncertainty. It relies on domain knowledge integrated with first-hand information that comes from data and generates solutions that are also accompanied by precise statements of reliability. In this paper, following recent developments in [22], we venture beyond the traditional set-up of scenario optimization by analyzing the concept of constraints relaxation. By a solid theoretical underpinning, this new paradigm furnishes fundamental tools to perform designs that meet a proper compromise between robustness and performance. After suitably expanding the scope of constraints relaxation as proposed in [22], we focus on various classical Support Vector methods in machine learning - including SVM (Support Vector Machine), SVR (Support Vector Regression) and SVDD (Support Vector Data Description) - and derive new results that attest the ability of these methods to generalize.