pr\u00e9sentent<\/p>\n\n\n\n
Autoris\u00e9 \u00e0 pr\u00e9senter ses travaux en vue de l\u2019obtention du Doctorat de l’Institut Polytechnique de Paris, pr\u00e9par\u00e9 \u00e0 T\u00e9l\u00e9com SudParis en :<\/p>\n\n\n\n
le MERCREDI 19 MARS 2025 \u00e0 14h00<\/p>\n\n\n\n
\u00e0<\/p>\n\n\n\n
Amphith\u00e9\u00e2tre 4
T\u00e9l\u00e9com Paris 19 Place Marguerite Perey 91120 Palaiseau<\/p>\n\n\n\n
Membres du jury :<\/strong><\/p>\n\n\n\n M. Randal DOUC<\/strong>, Professeur, T\u00e9l\u00e9com SudParis, FRANCE – Directeur de these R\u00e9sum\u00e9 :<\/strong><\/p>\n\n\n\n Cette th\u00e8se s’int\u00e9resse \u00e0 la conception et \u00e0 l’analyse d’algorithmes de cha\u00eenes de Markov s\u00e9quentielles et d’inf\u00e9rence variationnelle pour approcher des distributions complexes. La premi\u00e8re partie introduit une nouvelle classe de cha\u00eenes de Markov, appel\u00e9e Importance Markov chains. Il s’agit d’un m\u00e9ta-algorithme, construisant une cha\u00eene de Markov \u00e9tendue, visant une distribution cible sur sa premi\u00e8re composante, \u00e0 partir d’une cha\u00eene de Markov auxiliaire, visant une distribution instrumentale, par r\u00e9plication et rejet des points de la cha\u00eene auxiliaire. Cette cha\u00eene de Markov \u00e9tendue, conserve les propri\u00e9t\u00e9s de la cha\u00eene auxiliaire, \u00e0 savoir la loi des grands nombres (LGN), le th\u00e9or\u00e8me central limite (CLT) et l’ergodicit\u00e9 g\u00e9om\u00e9trique, sous des conditions faibles et quasiment optimales. La deuxi\u00e8me partie d\u00e9veloppe un algorithme s\u00e9quentiel de cha\u00eenes de Markov dites projet\u00e9es (pMC), obtenues comme la projection d’une cha\u00eene de Markov sur un sous-espace, permettant d’approcher une distribution cible fix\u00e9e (ex. tempering) ou une s\u00e9quence de distributions \u00e9voluant dans le temps pour des mod\u00e8les espace-\u00e9tat (ex. filtrage particulaire). Similairement aux m\u00e9thodes SMC, notre algorithme alterne entre phases de resampling et d’enrichissement-propagation, mais en maintenant des \u00e9chantillons pMC \u00e0 chaque \u00e9tape. En particulier, nous nous appuyons sur l’Importance Markov chain d\u00e9velopp\u00e9e dans la premi\u00e8re partie pour l’\u00e9tape de resampling, et sur des explorations \u00e0 l’aide d’un noyau de Markov pour l’enrichissement-propagation. Sous des hypoth\u00e8ses faibles, nous prouvons que la LGN, le CLT et surtout l’ergodicit\u00e9 g\u00e9om\u00e9trique se propagent \u00e0 travers les it\u00e9rations, constituant une am\u00e9lioration notable par rapport aux erreurs de resampling d’ordre O(1\/N) en SMC. Par ailleurs, les propri\u00e9t\u00e9s atomiques de l’Importance Markov chain permettent de parall\u00e9liser notre algorithme sur M n\u0153uds pour estimer des int\u00e9grales sous la loi cible, avec une erreur tendant vers 0 quand le nombre de n\u0153uds tend vers l’infini. La derni\u00e8re partie porte sur l’approximation param\u00e9trique de la cible par des m\u00e9thodes d’inf\u00e9rence variationnelle. Nous analysons notamment un algorithme d’inf\u00e9rence variationnelle par poids d’importance, c\u00e9l\u00e8bre pour son utilisation dans le cadre des auto-encodeurs variationnels, et prouvons des propri\u00e9t\u00e9s th\u00e9oriques (consistance, normalit\u00e9 asymptotique, efficience) jusqu’alors inexistantes dans la litt\u00e9rature. Abstract :<\/strong><\/p>\n\n\n\n This thesis focuses on the design and analysis of sequential Markov chain algorithms and variational inference to approximate complex distributions. The first part introduces a new class of Markov chains, called Importance Markov chains. This is a meta-algorithm that constructs an extended Markov chain, targeting a complex distribution on its first component, starting from an auxiliary Markov chain targeting an instrumental distribution, by replicating and rejecting points from the auxiliary chain. This extended Markov chain retains the properties of the auxiliary chain, namely the law of large numbers (LLN), the central limit theorem (CLT), and geometric ergodicity, under weak and nearly optimal conditions. The second chapter develops a sequential algorithm, based on so-called projected Markov chains (pMC), obtained by projecting a Markov chain onto a subspace. This approach allows the approximation of a fixed target distribution (e.g., tempering) or sequence of distributions evolving over time for state-space models (e.g., particle filtering). Similar to SMC methods, our algorithm alternates between resampling and enrichment-propagation phases but maintains pMC samples at each step. In particular, we rely on the Importance Markov chain developed in the first part for the resampling step and on explorations using a Markov kernel for enrichment-propagation. Under weak assumptions, we prove that LLN, CLT, and especially geometric ergodicity propagate through the iterations, constituting a significant improvement over the typical O(1\/N) error rate of resampling procedures in SMC. Furthermore, the atomic properties of the Importance Markov chain allow us to parallelize our algorithm for the empirical estimation of integrals under the target distribution, with an error tending toward 0 as the number of nodes increases. The last part focuses on the parametric approximation of a target distribution using variational inference. More specifically, we analyze a popular variational inference algorithm based on importance weighting, extensively used for variational autoencoders, and prove theoretical properties (consistency, asymptotic normality, efficiency) that were previously unavailable in the literature.<\/p>\n","protected":false},"excerpt":{"rendered":" L’Ecole doctorale : Math\u00e9matiques Hadamard et le Laboratoire de recherche SAMOVAR – Services r\u00e9partis, Architectures, Mod\u00e9lisation, Validation, Administration des R\u00e9seaux pr\u00e9sentent l\u2019AVIS DE SOUTENANCE de Monsieur Hugo MARIVAL Autoris\u00e9 \u00e0 pr\u00e9senter ses travaux en vue de l\u2019obtention du Doctorat de l’Institut Polytechnique de Paris, pr\u00e9par\u00e9 \u00e0 T\u00e9l\u00e9com SudParis en : \u00ab Algorithmes de cha\u00eenes de 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M. Victor ELVIRA<\/strong>, Professor, University of Edinburgh, ROYAUME-UNI – Rapporteur
M. Jean-Michel MARIN<\/strong>, Professeur, Universit\u00e9 de Montpellier, FRANCE – Rapporteur
M. Christian ROBERT<\/strong>, Professeur, Universit\u00e9 Paris-Dauphine, FRANCE – Examinateur
Mme H\u00e9l\u00e8ne HALCONRUY<\/strong>, Ma\u00eetresse de conf\u00e9rences, T\u00e9l\u00e9com SudParis, FRANCE – Examinateur
M. Julien STOEHR<\/strong>, Ma\u00eetre de conf\u00e9rences, Universit\u00e9 Paris-Dauphine, FRANCE – Examinateur<\/p>\n\n\n\n\u00ab Algorithmes de cha\u00eenes de Markov s\u00e9quentielles et d’inf\u00e9rence variationnelle : d\u00e9veloppement et analyse pour approcher des distributions complexes. \u00bb<\/h2>\n\n\n\n
pr\u00e9sent\u00e9 par Monsieur Hugo MARIVAL<\/h2>\n\n\n\n
<\/p>\n\n\n\n