Victor Léger

Abstract :
In the field of machine learning, the specific subfield of deep learning has gathered particular interest in the last decade. If deep learning has allowed quick and significant progress in a wide range of fields, this progress has been made at the expense of interpretability, accessibility, robustness and flexibility, not to mention the consecutive rebound effects of data center deployment and energy consumption implied by the training of such algorithms. In this context, the present manuscript aims on the contrary to open the path to tractable and flexible tools for classification problems, buttressed on elementary machine learning notions and rather basic mathematical tools. This thesis conducts a large dimensional study of a simple yet quite versatile classification model, encompassing at once multi-task and semi-supervised learning, and taking into account uncertain labeling. Using tools from random matrix theory, the asymptotics of some key functionals are characterized, which allows on the one hand to predict the performances of the proposed algorithm, and on the other hand to reveal some counter-intuitive guidance on how to use it efficiently. The model, powerful enough to provide good performance guarantees, is also straightforward enough to provide strong insight into its behavior. The resulting algorithm is also compared to an optimal bound derived from statistical physics, which gives a lower bound of the least achievable probability of misclassification for a given problem. This bound is computed in the extended case of uncertain labeling, and is used to evaluate the performances of the algorithm.