Le but de cette thèse est de montrer que l'arsenal des nouvelles méthodes d'optimisation permet de résoudre des problèmes d'estimation difficile basés sur les modèles d'évènements.Alors que le cadre classique de l'apprentissage supervisé traite les observations comme une collection de couples de covariables et de label, les modèles d'évènements ne regardent que les temps d'arrivée d'évènements et cherchent alors à extraire de l'information sur la source de donnée.Ces évènements datés sont ordonnés de façon chronologique et ne peuvent dès lors être considérés comme indépendants.Ce simple fait justifie l'usage d'un outil mathématique particulier appelé processus ponctuel pour apprendre une certaine structure à partir de ces évènements.Deux exemples de processus ponctuels sont étudiés dans cette thèse.Le premier est le processus ponctuel derrière le modèle de Cox à risques proportionnels:son intensité conditionnelle permet de définir le ratio de risque, une quantité fondamentale dans la littérature de l'analyse de survie.Le modèle de régression de Cox relie la durée avant l'apparition d'un évènement, appelé défaillance, aux covariables d'un individu.Ce modèle peut être reformulé à l'aide du cadre des processus ponctuels.Le second est le processus de Hawkes qui modélise l'impact des évènements passés sur la probabilité d'apparition d'évènements futurs.Le cas multivarié permet d'encoder une notion de causalité entre les différentes dimensions considérées.Cette thèse est divisée en trois parties.La première s'intéresse à un nouvel algorithme d'optimisation que nous avons développé.Il permet d'estimer le vecteur de paramètre de la régression de Cox lorsque le nombre d'observations est très important.Notre algorithme est basé sur l'algorithme SVRG (Stochastic Variance Reduced Gradient) et utilise une méthode MCMC (Monte Carlo Markov Chain) pour approcher un terme de la direction de descente.Nous avons prouvé des vitesses de convergence pour notre algorithme et avons montré sa performance numérique sur des jeux de données simulés et issus de monde réel.La deuxième partie montre que la causalité au sens de Hawkes peut être estimée de manière non-paramétrique grâce aux cumulants intégrés du processus ponctuel multivarié.Nous avons développer deux méthodes d'estimation des intégrales des noyaux du processus de Hawkes, sans faire d'hypothèse sur la forme de ces noyaux. Nos méthodes sont plus rapides et plus robustes, vis-à-vis de la forme des noyaux, par rapport à l'état de l'art. Nous avons démontré la consistence statistique de la première méthode, et avons montré que la deuxième peut être réduite à un problème d'optimisation convexe.La dernière partie met en lumière les dynamiques de carnet d'ordre grâce à la première méthode d'estimation non-paramétrique introduite dans la partie précédente.Nous avons utilisé des données du marché à terme EUREX, défini de nouveaux modèles de carnet d'ordre (basés sur les précédents travaux de Bacry et al.) et appliqué la méthode d'estimation sur ces processus ponctuels.Les résultats obtenus sont très satisfaisants et cohérents avec une analysé économétrique.Un tel travail prouve que la méthode que nous avons développé permet d'extraire une structure à partir de données aussi complexes que celles issues de la finance haute-fréquence. / The guiding principle of this thesis is to show how the arsenal of recent optimization methods can help solving challenging new estimation problems on events models.While the classical framework of supervised learning treat the observations as a collection of independent couples of features and labels, events models focus on arrival timestamps to extract information from the source of data.These timestamped events are chronologically ordered and can't be regarded as independent.This mere statement motivates the use of a particular mathematical object called point process to learn some patterns from events.Two examples of point process are treated in this thesis.The first is the point process behind Cox proportional hazards model:its conditional intensity function allows to define the hazard ratio, a fundamental quantity in survival analysis literature.The Cox regression model relates the duration before an event called failure to some covariates.This model can be reformulated in the framework of point processes.The second is the Hawkes process which models how past events increase the probability of future events.Its multivariate version enables encoding a notion of causality between the different nodes.The thesis is divided into three parts.The first focuses on a new optimization algorithm we developed to estimate the parameter vector of the Cox regression in the large-scale setting.Our algorithm is based on stochastic variance reduced gradient descent (SVRG) and uses Monte Carlo Markov Chain to estimate one costly term in the descent direction.We proved the convergence rates and showed its numerical performance on both simulated and real-world datasets.The second part shows how the Hawkes causality can be retrieved in a nonparametric fashion from the integrated cumulants of the multivariate point process.We designed two methods to estimate the integrals of the Hawkes kernels without any assumption on the shape of the kernel functions. Our methods are faster and more robust towards the shape of the kernels compared to state-of-the-art methods. We proved the statistical consistency of the first method, and designed turned the second into a convex optimization problem.The last part provides new insights from order book data using the first nonparametric method developed in the second part.We used data from the EUREX exchange, designed new order book model (based on the previous works of Bacry et al.) and ran the estimation method on these point processes.The results are very insightful and consistent with an econometric analysis.Such work is a proof of concept that our estimation method can be used on complex data like high-frequency financial data.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第21916号 / 情博第699号 / 新制||情||120(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)准教授 加嶋 健司, 教授 太田 快人, 教授 山下 信雄 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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Non-convex optimization problems appear in various applications of machine learning. Because of their practical importance, these problems gained a lot of attention in recent years, leading to the rapid development of new efficient stochastic gradient-type methods. In the quest to improve the generalization performance of modern deep learning models, practitioners are resorting to using larger and larger datasets in the training process, naturally distributed across a number of edge devices. However, with the increase of trainable data, the computational costs of gradient-type methods increase significantly. In addition, distributed methods almost invariably suffer from the so-called communication bottleneck: the cost of communication of the information necessary for the workers to jointly solve the problem is often very high, and it can be orders of magnitude higher than the cost of computation. This thesis provides a study of first-order stochastic methods addressing these issues. In particular, we structure this study by considering certain classes of methods. That allowed us to understand current theoretical gaps, which we successfully filled by providing new efficient algorithms.

<div>This work aims to provide solutions to two significant issues in the effective use and practical application of tensor completion as a machine learning method. The first solution addresses the challenge in designing fast and accurate recovery methods in tensor completion in the presence of highly sparse and highly missing data. The second takes on the need for robust uncertainty quantification methods for the recovered tensor.</div><div><br></div><div><b>Covariate-assisted Sparse Tensor Completion</b></div><div><b><br></b></div><div>In the first part of the dissertation, we aim to provably complete a sparse and highly missing tensor in the presence of covariate information along tensor modes. Our motivation originates from online advertising where users click-through-rates (CTR) on ads over various devices form a CTR tensor that can have up to 96% missing entries and has many zeros on non-missing entries. These features makes the standalone tensor completion method unsatisfactory. However, beside the CTR tensor, additional ad features or user characteristics are often available. We propose Covariate-assisted Sparse Tensor Completion (COSTCO) to incorporate covariate information in the recovery of the sparse tensor. The key idea is to jointly extract latent components from both the tensor and the covariate matrix to learn a synthetic representation. Theoretically, we derive the error bound for the recovered tensor components and explicitly quantify the improvements on both the reveal probability condition and the tensor recovery accuracy due to covariates. Finally, we apply COSTCO to an advertisement dataset from a major internet platform consisting of a CTR tensor and ad covariate matrix, leading to 23% accuracy improvement over the baseline methodology. An important by-product of our method is that clustering analysis on ad latent components from COSTCO reveal interesting and new ad clusters, that link different product industries which are not formed in existing clustering methods. Such findings could be directly useful for better ad planning procedures.</div><div><b><br></b></div><div><b>Uncertainty Quantification in Covariate-assisted Tensor Completion</b></div><div><br></div><div>In the second part of the dissertation, we propose a framework for uncertainty quantification for the imputed tensor factors obtained from completing a tensor with covariate information. We characterize the distribution of the non-convex estimator obtained from using the algorithm COSTCO down to fine scales. This distributional theory in turn allows us to construct proven valid and tight confidence intervals for the unseen tensor factors. The proposed inferential procedure enjoys several important features: (1) it is fully adaptive to noise heteroscedasticity, (2) it is data-driven and automatically adapts to unknown noise distributions and (3) in the high missing data regime, the inclusion of side information in the tensor completion model yields tighter confidence intervals compared to those obtained from standalone tensor completion methods.</div><div><br></div>

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This document presents the results of the master thesis conducted from April 2019 to October 2019 under the direction of CNES engineer Eric Bourgeois, as part of the KTH Master of Science in Aerospace Engineering curriculum. Within the framework of development studies for the Callisto demonstrator, this master thesis aims at studying and developing possible guidance strategies for the boosted landing. Two main approaches are described in this document : • Adaptive pseudo-spectral interpolation • Convex optimization The satisfying results yielded give strong arguments for choosing the latter as part of the Callisto GNC systems and describe possible implementation strategies as well as complementary analyses that could be conducted. / Denna rapport presenterar resultaten av ett examensarbete som genomfördes från april till oktober 2019 under ledning av CNES-ingenjören Eric Bourgeois, som en del av en masterexamen i flyg- och rymdteknik från KTH, Kungliga tekniska högskolan. Inom ramen för utvecklingsstudier för bärraketen Callisto syftar detta arbete att studera och utveckla möjliga reglerstrategier för Callistos landing som kontrolleras med raketer. Två huvudsakliga metoder beskrivs: • Adaptiv pseudospektral interpolering • Konvex optimering Resultaten ger starka argument för att välja den senare av dessa två metoder för Callistos reglersystem och beskriver möjliga implementeringsstrategier samt vilka kompletterande analyser som bör genomföras

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