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 l’Institut Polytechnique de Paris T\u00e9l\u00e9com SudParis en :<\/p>\n\n\n\n
le MARDI 13 OCTOBRE 2026 \u00e0 14h00<\/p>\n\n\n\n
\u00e0<\/p>\n\n\n\n
Amphith\u00e9\u00e2tre 2
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, Institut Polytechnique de Paris T\u00e9l\u00e9com SudParis, FRANCE – Directeur de th\u00e8se Invit\u00e9 : <\/strong><\/p>\n\n\n\n M. Fran\u00e7ois ROUEFF<\/strong>, Professeur, Institut Polytechnique de Paris T\u00e9l\u00e9com Paris, FRANCE – Co-directeur de th\u00e8se<\/p>\n\n\n\n R\u00e9sum\u00e9 :<\/strong><\/p>\n\n\n\n De nombreuses questions en statistique, en apprentissage automatique et en sciences physiques se ram\u00e8nent au calcul d’esp\u00e9rances pour lesquelles il n’existe pas de forme close. Ce d\u00e9fi se pose en particulier dans deux grandes classes de probl\u00e8mes : l’inf\u00e9rence bay\u00e9sienne, o\u00f9 la distribution cible est connue \u00e0 une constante de normalisation pr\u00e8s, et la mod\u00e9lisation g\u00e9n\u00e9rative, o\u00f9 elle n’est accessible qu’au travers d’\u00e9chantillons. Cette th\u00e8se d\u00e9veloppe th\u00e9orie et algorithmes pour trois strat\u00e9gies compl\u00e9mentaires permettant de relever ce d\u00e9fi : l’inf\u00e9rence variationnelle, la mod\u00e9lisation g\u00e9n\u00e9rative, et les m\u00e9thodes de Monte Carlo par cha\u00eenes de Markov. La premi\u00e8re partie pose le cadre des probl\u00e8mes et introduit des outils importants qui serviront tout au long du manuscrit. En particulier, nous explorons certaines propri\u00e9t\u00e9s fondamentales des familles exponentielles. La deuxi\u00e8me partie \u00e9tudie des m\u00e9thodes de minimisation d’alpha-divergences dans le cas o\u00f9 la famille variationnelle est un mod\u00e8le exponentiel. L’algorithme central est une proc\u00e9dure it\u00e9rative dont le but est de faire correspondre les moments de la distribution variationnelle et d’une cible particuli\u00e8re. Nous \u00e9tablissons une analyse d\u00e9taill\u00e9e de la convergence de l’AM dans un cadre d\u00e9terministe, et nous proposons ensuite deux extensions au cadre stochastique. Dans le contexte de l’approximation stochastique, nous introduisons une variante utilisant des estimateurs sans biais. Nous d\u00e9veloppons \u00e9galement un cadre d’Approximation par Moyenne Empirique, qui fixe des \u00e9chantillons au d\u00e9but de la proc\u00e9dure d’optimisation. La troisi\u00e8me partie aborde la mod\u00e9lisation g\u00e9n\u00e9rative sous l’angle de l’inf\u00e9rence variationnelle. Nous rappelons d’abord l’\u00e9quivalence formelle entre les auto-encodeurs variationnels hi\u00e9rarchiques (HVAEs) et les mod\u00e8les de diffusion, en mobilisant divers arguments de la litt\u00e9rature existante. Nous apportons une validation empirique originale des r\u00e9sultats th\u00e9oriques sur plusieurs jeux de donn\u00e9es. Nous introduisons ensuite M-Star, une m\u00e9thode de diffusion permettant d’utiliser des distributions de familles exponentielles dans le processus de bruitage. Les m\u00e9thodes pr\u00e9c\u00e9demment propos\u00e9es pour d\u00e9passer la limite des mod\u00e8les se basant sur des bruits gaussiens n\u00e9cessitaient de nouveaux calculs pour chaque type de distribution, ou pr\u00e9sentaient des instabilit\u00e9s num\u00e9riques. Nous proposons d’apprendre une projection markovienne d’un processus de bruitage non-markovien, con\u00e7u pour un bruit non-gaussien. Par sa g\u00e9n\u00e9ralit\u00e9, la m\u00e9thode est applicable \u00e0 des g\u00e9om\u00e9tries contraintes et \u00e0 des donn\u00e9es directionnelles, des situations dans lesquelles il ne serait pas viable d’utiliser des distributions gaussiennes. Une riche \u00e9tude exp\u00e9rimentale confirme la validit\u00e9 de la m\u00e9thode propos\u00e9e. La quatri\u00e8me partie compare des m\u00e9thodes de Monte-Carlo par cha\u00eene de Markov utilisant des propositions ind\u00e9pendantes. Nous \u00e9tudions trois noyaux \u00e0 essais multiples, le Multiple-Try Metropolis with Independent Balancing (MIB), le Multiple-Try Metropolis with Independent Sampling (MIS) et l’Iterated Sampling Importance Resampling (ISIR), et \u00e9tablissons une hi\u00e9rarchie th\u00e9orique stricte entre eux au sens de l’ordre de Peskun. Nous montrons en particulier que MIS domine \u00e0 la fois MIB et ISIR. Nous obtenons des vitesses de convergence exactes pour chacun de ces noyaux lorsque les poids d’importance sont born\u00e9s, et montrons qu’ils ne peuvent jamais \u00eatre g\u00e9om\u00e9triquement ergodiques lorsque ces poids sont non born\u00e9s. Or, cette situation correspond au r\u00e9gime pratique le plus courant. Cette limitation motive l’\u00e9tude de l’Independent Importance Markov Chain (IIMC), qui repose sur une structure de renouvellement dans un espace d’\u00e9tats augment\u00e9. Pour ce noyau, nous prouvons qu’une simple condition de moment exponentiel suffit \u00e0 garantir l’ergodicit\u00e9 g\u00e9om\u00e9trique, tandis que l’ergodicit\u00e9 polynomiale repose sur une condition de moment polynomial pour les poids d’importance. Many problems in statistics, machine learning, and the physical sciences reduce to computing expectations under intractable probability distributions. This thesis develops theory and algorithms for three complementary strategies to tackle this challenge: variational inference, generative modeling, and Markov Chain Monte-Carlo. The first part frames the problems and introduces important tools that will be used throughout the manuscript, in particular we focus on some properties of exponential families. The second part studies alpha-divergence minimization methods over exponential family variational distributions. The central algorithm is an iterative moment-matching scheme. We establish a detailed convergence analysis of the MA in a deterministic setting, and we propose two extensions to the stochastic setting. In the context of Stochastic Approximation, we introduce an unbiased stochastic variant. We also develop a Sample Average Approximation framework. The third part turns to generative modeling through the lens of variational inference. We first recall the formal equivalence between Hierarchical Variational Auto-Encoders (HVAEs) and diffusion models, invoking various arguments found in the existing literature to show that diffusion models arise as the infinite-depth limit of properly parameterized HVAEs. We provide novel empirical validation across several benchmark datasets. We then introduce M-Star, a diffusion framework that allows the noising process to use general exponential family distributions. The key idea is to learn a Markovian projection of a non-Markovian forward process designed for non-Gaussian noise, which makes the framework applicable to constrained geometries and directional data without requiring bespoke derivations for each target distribution. Extensive experiments confirm the validity of this new method. The fourth part compares Monte-Carlo methods based on independent proposals. We study three multiple-try kernels, the Multiple-Try Metropolis with Independent Balancing (MIB), the Multiple-Try Metropolis with Independent Sampling (MIS), and Iterated Sampling Importance Resampling (ISIR); and establish theoretical strict hierarchy among them via the Peskun ordering. Specifically, we show that MIS dominates both MIB and ISIR. We derive exact convergence rates for all three kernels when importance weights are bounded, and show that they fail to be geometrically ergodic when weights are unbounded. This limitation motivates the study of the Independent Importance Markov Chain, for which we prove geometric ergodicity under an exponential moment condition, and polynomial ergodicity under a polynomial moment condition. The results of this thesis are supported by publications at NeurIPS 2024, AISTATS 2026, ICLR 2026, and ICML 2026, as well as an unpublished paper currently under review.<\/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 Fran\u00e7ois BERTHOLOM Autoris\u00e9 \u00e0 pr\u00e9senter ses travaux en vue de l\u2019obtention du Doctorat de l’Institut Polytechnique de Paris, pr\u00e9par\u00e9 \u00e0 l’Institut Polytechnique de Paris T\u00e9l\u00e9com SudParis en : \u00ab 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M. Victor ELVIRA<\/strong>, Professor, School of Mathematics, University of Edinburgh, ROYAUME-UNI – Rapporteur
M. Julyan ARBEL<\/strong>, Charg\u00e9 de recherche, INRIA Grenoble, FRANCE – Rapporteur
Mme C\u00e9line L\u00e9VY-LEDUC<\/strong>, Professeure des universit\u00e9s, Universit\u00e9 Paris Cit\u00e9 (UFR de math\u00e9matiques), FRANCE – Examinateur
M. Nicolas CHOPIN<\/strong>, Professeur, ENSAE, FRANCE – Examinateur<\/p>\n\n\n\n\u00ab Th\u00e9orie et Algorithmes d\u2019Inf\u00e9rence Approch\u00e9e avec des Familles Exponentielles \u00bb<\/h2>\n\n\n\n
pr\u00e9sent\u00e9 par Monsieur Fran\u00e7ois BERTHOLOM<\/h2>\n\n\n\n
Abstract :<\/strong><\/p>\n\n\n\n