




Thèse de DOCTORAT Nonsmooth Optimization for Statistical Learning with Structured Matrix Regularization Training machine learning methods boils down to solving optimization problems whose objective functions often decomposes into two parts: a) the empirical risk, built upon the loss function, whose shape is determined by the performance metric and the noise assumptions; b) the regularization penalty, built upon a norm, or a gauge function, whose structure is determined by the prior information available for the problem at hand. Common loss functions, such as the hinge loss for binary classification, or more advanced loss functions, such as the one arising in classification with reject option, are nonsmooth. Sparse regularization penalties such as the (vector) L1penalty, or the (matrix) nuclearnorm penalty, are also nonsmooth. The goal of this thesis is to study doubly nonsmooth learning problems (with nonsmooth loss functions and nonsmooth regularization penalties) and firstorder optimization algorithms that leverage the composite structure of nonsmooth objectives. In the first chapter, we introduce new regularization penalties, called the group Schatten norms, to generalize the standard Schatten norms to blockstructured matrices. We establish the main properties of the group Schatten norms using tools from convex analysis and linear algebra; we retrieve in particular some convex envelope properties. We discuss several potential applications of the group nuclearnorm, in collaborative filtering, database compression, multilabel image tagging. In the second chapter, we present a survey of smoothing techniques that allow us to use firstorder optimization algorithms originally designed for learning problems with nonsmooth loss. We also show how smoothing can be used on the loss function corresponding to the topk accuracy, used for ranking and multiclass classification problems. We outline some firstorder algorithms that can be used in combination with the smoothing technique: i) conditional gradient algorithms; ii) proximal gradient algorithms; iii) incremental gradient algorithms. In the third chapter, we study further conditional gradient algorithms for solving doubly nonsmooth optimization problems. We show that an adaptive smoothing combined with the standard conditional gradient algorithm gives birth to new conditional gradient algorithms having the expected theoretical convergence guarantees. We present promising experimental results in collaborative filtering for movie recommendation and image categorization. MotsClés: fi rstorder optimization, conditional gradient, smoothing, nuclearnorm, machine learning, mathematical optimization Membres du Jury: Mr MassihReza AMINI (Université Grenoble Alpes), proposé président. Rapporteurs: Mr Alexander NAZIN (Institute of Control Sciences RAS, Moscow, Russia) Mr Stephane CHRETIEN (National Physical Laboratory, Teddington, Middlesex, UK) Examinateurs: Mme Nelly PUSTELNIK (CNRS, ENS Lyon) Mr Joseph SALMON (TELECOM ParisTech, Paris, France,) 

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