Random monotone operators and application to stochastic optimization

Abstract : This thesis mainly studies optimization algorithms. Programming problems arising in signal processing and machine learning are composite in many cases, i.e they exhibit constraints and non smooth regularization terms. Proximal methods are known to be efficient to solve such problems. However, in modern applications of data sciences, functions to be minimized are often represented as statistical expectations, whose evaluation is intractable. This cover the case of online learning, big data problems and distributed computation problems. To solve this problems, we study in this thesis proximal stochastic methods, that generalize proximal algorithms to the case of cost functions written as expectations. Stochastic proximal methods are first studied with a constant step size, using stochastic approximation techniques. More precisely, the Ordinary Differential Equation method is adapted to the case of differential inclusions. In order to study the asymptotic behavior of the algorithms, the stability of the sequences of iterates (seen as Markov chains) is studied. Then, generalizations of the stochastic proximal gradient algorithm with decreasing step sizes are designed to solve composite problems. Every quantities used to define the optimization problem are written as expectations. This include a primal dual algorithm to solve regularized and linearly constrained problems and an optimization over large graphs algorithm.
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Adil Salim. Random monotone operators and application to stochastic optimization. Optimization and Control [math.OC]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLT021⟩. ⟨tel-01960496⟩

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