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Un des thèmes pratiques, mais hautement sensibles, est le problème de sondage dans les réseaux sociaux où le caractère secret des informations sélectionnées et la réputation de l’utilisateur sont très critiques. En effet, les utilisateurs désirent préserver la confidentialité de leurs votes et dissimuler, le cas échéant, leurs mauvais comportements. Récemment, Guerraoui et al. ont propose des protocoles de sondage basés sur la partage de secret et ne nécessitant aucune infrastructure cryptographique. Néanmoins, ces protocoles ne sont applicables que si le graphe social a une structure d’anneau et le nombre d’utilisateurs est un carré parfait. Dans cette thèse, nous traitons, d’une part, du problème du déploiement décentralisé des protocoles de sondage qui sont basés sur des graphes sociaux ayant des structures générales, et d’autre part, du problème de transformation des graphes sociaux pour augmenter les propriétés de vie privée et de précision, nécessaires au déroulement sûr et rentable du sondage décentralisé. Premièrement, nous proposons trois protocoles décentralisés qui s’appuient sur l’état originel (sans transformation) des graphes sociaux. Les deux premiers protocoles utilisent respectivement des modèles de communication synchrone et asynchrone, et manipulent des procédures de vérification pour détecter les utilisateurs malhonnêtes. Quant au troisième protocole, il est asynchrone et ne nécessite pas de procédures de vérification. Pour que ce protocole permette une diffusion efficace de messages, nous avons défini une propriété sur les graphes sociaux, appelée “m-broadcasting”. Dans la deuxième partie de la thèse, nous formalisons le problème de “l’ajout des amis” qui consiste à trouver une transformation optimale des graphes sociaux pour les adapter au partage de secret. Pour résoudre ce problème, nous présentons deux algorithmes selon deux approches différentes: centralisée et décentralisée. Une évaluation expérimentale montre que nos protocoles sont précis et restreints aux bornes théoriques / One of the current practical, useful but sensitive topic in social networks is polling problem where the privacy of exchanged information and user reputation are very critical. Indeed, users want to preserve the confidentiality of their votes and to hide, if any, their misbehaviors. Recently, Guerraoui et al. proposed polling protocols based on simple secret sharing scheme and without requiring any central authority or cryptography system. However these protocols can be deployed safely and efficiently provided that the social graph structure should be transformed into a ring structure-based overlay and the number of participating users is perfect square. In this thesis, we address the problem of deploying decentralized polling protocols for general social graphs and how to transform these graphs in order to increase the privacy and/or accuracy properties. First, we propose three simple decentralized polling protocols that rely on the current state of social graphs. The two first protocols use synchronous and asynchronous models and verification procedures to detect the misbehaving users. The third protocol is an asynchronous one that does not require any verification procedures and contains a method for efficiently broadcasting message under a family of social graphs satisfying what we call the “m-broadcasting” property. Second, we formalize the “adding friends” problem such that we can reuse the social graphs after some minimum structural modifications consisting in adding new friendship relations. We also devise algorithms for solving this problem in centralized and decentralized networks. We validate our solutions with some performance evaluations which show that our protocols are accurate, and inside the theoretical bounds

<p>Recent decades have witnessed the rise of data deluge generated by heterogeneous sources, e.g., social networks, streaming, marketing services etc., which has naturally created a surge of interests in theory and applications of large-scale convex and non-convex optimization. For example, real-world instances of statistical learning problems such as deep learning, recommendation systems, etc. can generate sheer volumes of spatially/temporally diverse data (up to Petabytes of data in commercial applications) with millions of decision variables to be optimized. Such problems are often referred to as Big-data problems. Solving these problems by standard optimization methods demands intractable amount of centralized storage and computational resources which is infeasible and is the foremost purpose of parallel and decentralized algorithms developed in this thesis.</p><p><br></p><p>This thesis consists of two parts: (I) Distributed Nonconvex Optimization and (II) Distributed Convex Optimization.</p><p><br></p><p>In Part (I), we start by studying a winning paradigm in big-data optimization, Block Coordinate Descent (BCD) algorithm, which cease to be effective when problem dimensions grow overwhelmingly. In particular, we considered a general family of constrained non-convex composite large-scale problems defined on multicore computing machines equipped with shared memory. We design a hybrid deterministic/random parallel algorithm to efficiently solve such problems combining synergically Successive Convex Approximation (SCA) with greedy/random dimensionality reduction techniques. We provide theoretical and empirical results showing efficacy of the proposed scheme in face of huge-scale problems. The next step is to broaden the network setting to general mesh networks modeled as directed graphs, and propose a class of gradient-tracking based algorithms with global convergence guarantees to critical points of the problem. We further explore the geometry of the landscape of the non-convex problems to establish second-order guarantees and strengthen our convergence to local optimal solutions results to global optimal solutions for a wide range of Machine Learning problems.</p><p><br></p><p>In Part (II), we focus on a family of distributed convex optimization problems defined over meshed networks. Relevant state-of-the-art algorithms often consider limited problem settings with pessimistic communication complexities with respect to the complexity of their centralized variants, which raises an important question: can one achieve the rate of centralized first-order methods over networks, and moreover, can one improve upon their communication costs by using higher-order local solvers? To answer these questions, we proposed an algorithm that utilizes surrogate objective functions in local solvers (hence going beyond first-order realms, such as proximal-gradient) coupled with a perturbed (push-sum) consensus mechanism that aims to track locally the gradient of the central objective function. The algorithm is proved to match the convergence rate of its centralized counterparts, up to multiplying network factors. When considering in particular, Empirical Risk Minimization (ERM) problems with statistically homogeneous data across the agents, our algorithm employing high-order surrogates provably achieves faster rates than what is achievable by first-order methods. Such improvements are made without exchanging any Hessian matrices over the network. </p><p><br></p><p>Finally, we focus on the ill-conditioning issue impacting the efficiency of decentralized first-order methods over networks which rendered them impractical both in terms of computation and communication cost. A natural solution is to develop distributed second-order methods, but their requisite for Hessian information incurs substantial communication overheads on the network. To work around such exorbitant communication costs, we propose a “statistically informed” preconditioned cubic regularized Newton method which provably improves upon the rates of first-order methods. The proposed scheme does not require communication of Hessian information in the network, and yet, achieves the iteration complexity of centralized second-order methods up to the statistical precision. In addition, (second-order) approximate nature of the utilized surrogate functions, improves upon the per-iteration computational cost of our earlier proposed scheme in this setting.</p>