Mathematics, Machine Learning, Sciences, and Humanities - Master's Year 2

Cours introductifs

• Introduction to R

  Introduction to R

  Lecturer :
  STANISLAS LEULIER DU CHE
  

  Total hours : 3

  Overview :

  Introduction to the R programming language: loading data, writing simple functions, producing standard plots.

  Learning outcomes :

  Programming in R

  Assessment :

  No evaluation

• Introduction to Bayesian Statistics

  Introduction to Bayesian Statistics

  Lecturer :
  CHRISTIAN ROBERT
  

  Total hours : 3
• A review of probability theory foundations

  A review of probability theory foundations

  Lecturer :
  PAUL GASSIAT
  

  Total hours : 15

  Overview :

  Outline :

  1. Basics of measure theory and integration

  2. Probability : random variables, independence

  3. Convergence of random variables

  4. Law of Large Numbers and Central Limit Theorem

  5. Conditional expectations

  6. Martingales in discrete time

  7. Gaussian vectors

  8. Brownian motion : definition, existence, first properties

  Learning outcomes :

  The aim of this class is to provide a quick review of the probability theory that is required to follow the 1st semester classes in MATH, MASEF and MASH.

  Most of the content should already be familiar to students with a M1 in Mathematics.

• Introduction to Python

  Introduction to Python

  Lecturer :
  David GONTIER
  

  Total hours : 3

  Overview :

  Dans ce cours de 3h, nous voyons (ou re-voyons) la base de Python, et l'utilisation des notebooks. Il est illustré par 3 notebooks. Le premier rappelle les bases générales de Python. Le second porte sur l'utilisation du module pandas, et le dernier sur un problème simple d'optimisation de portfolio.

  Learning outcomes :

  • Installer Python sur sa machine
  • Utiliser un notebook
  • Savoir lire la documentation de Python, et écrire des codes simples

Cours fondamentaux

• Optimization for Machine Learning

  Optimization for Machine Learning

  Ects : 6

  Lecturer :
  GABRIEL PEYRE
  IRENE WALDSPURGER
  CLEMENT ROYER
  

  Total hours : 48

  Overview :

  This course delves into the mathematical underpinnings and algorithmic strategies essential for understanding and applying Machine Learning techniques. Central to the course is the exploration of optimization, a pivotal element in contemporary advancements in machine learning. This exploration encompasses fundamental approaches such as linear regression, SVMs, and kernel methods, and extends to the dynamic realm of deep learning. Deep learning has become a leading methodology for addressing a variety of challenges in areas like imaging, vision, and natural language processing. The course content is structured to provide a comprehensive overview of the mathematical foundations, algorithmic methods, and a variety of modern applications utilizing diverse optimization techniques. Participants will engage in both traditional lectures and practical numerical sessions using Python. The curriculum is divided into three parts: The first focuses on smooth and convex optimization techniques, including gradient descent and duality. The second part delves into advanced methods like non-smooth optimization and proximal methods. Lastly, the third part addresses large-scale methods such as stochastic gradient descent and automatic differentiation, with a special focus on their applications in neural networks, including both shallow and deep architectures.

   

  Detailed Syllabus:

  1. Foundational Concepts in Differential Calculus and Gradient Descent: - Introduction to differential calculus - Principles of gradient descent - Application of gradient descent in optimization

  2. Automatic Differentiation and Its Applications: - Understanding the mechanics of automatic differentiation - Implementing automatic differentiation using modern Python frameworks

  3. Advanced Gradient Descent Techniques: - In-depth study of gradient descent theory - Accelerated gradient methods - Stochastic gradient algorithms and their applications

  4. Exploring Convex and Non-Convex Optimization: - Fundamentals of convex analysis - Strategies and challenges in non-convex optimization

  5. Special Topics in Optimization: - Introduction to non-smooth optimization methods - Study of semidefinite programming (SDP) - Exploring interior points and proximal methods

  6. Large-Scale Optimization Methods and Neural Networks: - Techniques in large-scale methods, focusing on stochastic gradient descent - Applications of automatic differentiation in neural networks - Overview of neural network architectures: shallow and deep networks  

  Bibliography-recommended reading

  • Theory and algorithms: Convex Optimization, Boyd and Vandenberghe
  • Introduction to matrix numerical analysis and optimization, Philippe Ciarlet
  • Proximal algorithms, N. Parikh and S. Boyd
  • Introduction to Nonlinear Optimization - Theory, Algorithms and Applications, Amir Beck
  • Numerics: Python and Jupyter installation: use only Python 3 with Anaconda distribution.
  • The Numerical Tours of Signal Processing, Gabriel Peyré
  • Scikitlearn tutorial, Fabian Pedregosa, Jake VanderPlas
• High-dimensional statistics

  High-dimensional statistics

  Ects : 5

  Lecturer :
  VINCENT RIVOIRARD
  

  Total hours : 24

  Overview :

  Fléau de la dimension et hypothèse de parcimonie pour la régression gaussienne, les modèles linéaires généralisés et les données de comptage.

  Ondelettes et estimation par seuillage.

  Choix de modèles et sélection de variables.

  Estimation par pénalisation convexe : procédure Ridge, lasso, group-lasso… Liens avec l’approche bayésienne.

  Tests multiples : procédures FDR, FWER.

  Données fonctionnelles

  Recommended prerequisites :

  Bon niveau de statistique mathématique de niveau M1

  Require prerequisites :

  M1 de mathématiques appliquées

  Learning outcomes :

  L’objectif de ce cours de statistique est de présenter les outils mathématiques et les méthodologies dans la situation où le nombre de paramètres à inférer est très élevé, typiquement beaucoup plus important que le nombre d’observations.

  Assessment :

  Examen sur table

  Learn more about the course :

  www.ceremade.dauphine.fr/~rivoirar/Poly-Mash.pdf

  Bibliography-recommended reading

  Wasserman, L. (2005) All of statistics. A concise course in statistical inference. Springer

• Foundations of machine learning

  Foundations of machine learning

  Ects : 5

  Lecturer :
  FRANCIS BACH
  

  Total hours : 24

  Overview :

  Typologie des problèmes d ’ apprentissage (supervisé vs. non-supervisé). Modèle statistique pour la classification binaire : Approches génératives vs. discriminantes. Algorithmes classiques : méthodes paramétriques, perceptron, méthodes de partitionnement. Critères de performances : erreur de classification, courbe ROC, AUC. Convexification du risque : Algorithmes de type boosting et SVM. Mesures de complexité combinatoires, métriques géométriques. Sélection de modèle et régularisation. Théorèmes de consistance et vitesses de convergence.

  Learning outcomes :

  Bases mathématiques pour la modélisation des problèmes d’apprentissage supervisé et l’analyse des algorithmes de classification en grande dimension. Il s’agit de présenter les bases mathématiques pour la modélisation des problèmes d’apprentissage supervisé et l’analyse des algorithmes de classification en grande dimension.

• Graphical models

  Graphical models

  Ects : 4

  Lecturer :
  FABRICE ROSSI
  

  Total hours : 18

  Learning outcomes :

  Modélisation probabiliste, apprentissage et inférence sur les modèles graphiques. Les principaux thèmes abordés sont :Maximum de vraisemblance.Régression linéaire.Régression logistique.Modèle de mélange, partitionnement.Modèles graphiques.Familles exponentielles.Algorithme produit-somme.Hidden Markov models.Inférence approximéeMéthodes bayésiennes.

Cours optionnels - 5 cours à choisir parmi :

• Optimal transport

  Optimal transport

  Ects : 4

  Lecturer :
  GABRIEL PEYRE
  

  Total hours : 18

  Overview :

  Optimal transport (OT) is a fundamental mathematical theory at the interface between optimization, partial differential equations and probability. It has recently emerged as an important tool to tackle a surprisingly large range of problems in data sciences, such as shape registration in medical imaging, structured prediction problems in supervised learning and training deep generative networks. This course will interleave the description of the mathematical theory with the recent developments of scalable numerical solvers. This will highlight the importance of recent advances in regularized approaches for OT which allow one to tackle high dimensional learning problems.

  The course will feature numerical sessions using Python.

  • Motivations, basics of probabilistic modeling and matching problems.
  • Monge problem, 1D case, Gaussian distributions.
  • Kantorovitch formulation, linear programming, metric properties.
  • Shrödinger problem, Sinkhorn algorithm.
  • Duality and c-transforms, Brenier’s theory, W1, generative modeling.
  • Semi-discrete OT, quantization, Sinkhorn dual and divergences
• Computational methods and MCMC

  Computational methods and MCMC

  Ects : 4

  Lecturer :
  CHRISTIAN ROBERT
  

  Total hours : 21

  Overview :

  Motivations Monte-Carlo Methods Markov Chain Reminders The Metropolis-Hastings method The Gibbs Sampler Perfect sampling Sequential Monte-Carlo methods

  Learning outcomes :

  This course aims at presenting the basics and recent developments of simulation methods used in statistics and especially in Bayesian statistics. Methods of computation, maximization and high-dimensional integration have indeed become necessary to deal with the complex models envisaged in the user disciplines of statistics, such as econometrics, finance, genetics, ecology or epidemiology (among others!). The main innovation of the last ten years is the introduction of Markovian techniques for the approximation of probability laws (and the corresponding integrals). It thus forms the central part of the course, but we will also deal with particle systems and stochastic optimization methods such as simulated annealing.

• Applied Bayesian statistics

  Applied Bayesian statistics

  Ects : 4

  Lecturer :
  ROBIN RYDER
  

  Total hours : 18

  Overview :

  We shall put in practice classical models for statistical inference in a Bayesian setting, and implement computational methods. Using real data, we shall study various models such as linear regression, capture-recapture, and a hierarchical model. We shall discuss issues of model building and validation, the impact of the choice of prior, and model choice via Bayes Factors. The implementation shall use several algorithms: Markov Chain Monte Carlo, importance sampling, Approximate Bayesian Computation. The course is based on the free software R. Practical information: Large portions of the course are devoting to students coding. Students should bring their own laptop, which must have R installed before the first session; I strongly suggest installing RStudio (free) as well.

  Require prerequisites :

  Pre-requisite: Knowledge of the programming language R is essential, as well as an introduction to Bayesian inference.

  Learning outcomes :

  Modelling and inference in a Bayesian setting

• Bayesian non parametric and Bayesian Machine Learning

  Bayesian non parametric and Bayesian Machine Learning

  Ects : 4

  Lecturer :
  GUILLAUME KON KAM KING
  

  Total hours : 18

  Overview :

  Bayesian nonparametrics:

  • Basics: infinite mixture models and clustering
  • Models beyond the Dirichlet process
  • Posterior sampling
  • Applications

  Gaussian Processes

  Bayesian Deep Learning

  Require prerequisites :

  Bayesian statistics

  Learning outcomes :

  Essential concepts of Bayesian nonparametrics

  Essentials of Bayesian Deep Learning

  Bibliography-recommended reading

  • Hjort, N. L., Holmes, C., Müller, P., & Walker, S. G. (Eds.). (2010). Bayesian nonparametrics (Vol. 28). Cambridge University Press.
  • Orbanz, P., & Teh, Y. W. (2010). Bayesian nonparametric models. Encyclopedia of machine learning, 1, 81-89.
  • Müller, P., Quintana, F. A., Jara, A., & Hanson, T. (2015). Bayesian nonparametric data analysis (Vol. 1). New York: Springer.
  • Ghosal, S., & van der Vaart, A. W. (2017). Fundamentals of nonparametric Bayesian inference (Vol. 44). Cambridge University Press.
  • Murphy, K. P. (2012). Machine learning: a probabilistic perspective. MIT press.
  • Murphy, K. P. (2023). Probabilistic machine learning: Advanced topics. MIT press.
• Mixing times of Markov chains

  Mixing times of Markov chains

  Ects : 4

  Lecturer :
  JUSTIN SALEZ
  

  Total hours : 24

  Overview :

  How many times must one shuffle a deck of 52 cards? This course is a self-contained introduction to the modern theory of mixing times of Markov chains. It consists of a guided tour through the various methods for estimating mixing times, including couplings, spectral analysis, discrete geometry, and functional inequalities. Each of those tools is illustrated on a variety of examples from different contexts: interacting particle systems, card shuffling, random walks on groups, graphs and networks, etc. Finally, a particular attention is devoted to the celebrated cutoff phenomenon, a remarkable but still mysterious phase transition in the convergence to equilibrium of certain Markov chains.

  Learning outcomes :

  See the webpage of the course.

  Assessment :

  Final written exam, in class.

  Learn more about the course :

  www.ceremade.dauphine.fr/~salez/mix.html

  Bibliography-recommended reading

  See the webpage of the course.

• Large Language Models

  Large Language Models

  Ects : 4

  Lecturer :
  ALEXANDRE ALLAUZEN
  Florian LE BRONNEC
  

  Total hours : 24

  Overview :

  The course focuses on modern and statistical approaches to NLP.

  Natural language processing (NLP) is today present in some many applications because people communicate most everything in language : post on social media, web search, advertisement, emails and SMS, customer service exchange, language translation, etc. While NLP heavily relies on machine learning approaches and the use of large corpora, the peculiarities and diversity of language data imply dedicated models to efficiently process linguistic information and the underlying computational properties of natural languages.

  Moreover, NLP is a fast evolving domain, in which cutting-edge research can nowadays be introduced in large scale applications in a couple of years.

  The course focuses on modern and statistical approaches to NLP: using large corpora, statistical models for acquisition, disambiguation, parsing, understanding and translation. An important part will be dedicated to deep-learning models for NLP.

  - Introduction to NLP, the main tasks, issues and peculiarities - Sequence tagging: models and applications - Computational Semantics - Syntax and Parsing - Deep Learning for NLP: introduction and basics - Deep Learning for NLP: advanced architectures - Deep Learning for NLP: Machine translation, a case study

  Recommended prerequisites :

  pytorch

  Learning outcomes :

  • Skills in Natural Language Processing using deep-learning
  • Understand new architectures
• Reinforcement Learning

  Reinforcement Learning

  Ects : 4

  Lecturer :
  ANA BUSIC
  

  Total hours : 24
• Kernel methods

  Kernel methods

  Ects : 4

  Lecturer :
  MICHAEL ARBEL
  

  Total hours : 18

  Overview :

  Reproducing kernel Hilbert spaces et le “kernel trick”Théorème de représentationKernel PCAKernel ridge regressionSupport vector machinesNoyaux sur les semigroupesNoyaux pour le texte, les graphes, etc.

  Learning outcomes :

  Présenter les bases théoriques et des applications des méthodes à noyaux en apprentissage.

• Non-convex inverse problems

  Non-convex inverse problems

  Ects : 4

  Lecturer :
  IRENE WALDSPURGER
  

  Total hours : 18

  Overview :

  An inverse problem is a problem where the goal is to recover an unknown object (typically a vector with real coordinates, or a matrix), given a few ``measurements'' of this object, and possibly some information on its structure. In this course, we will discuss examples of such problems, motivated by applications as diverse as medical imaging, optics and machine learning. We will especially focus on the questions: which algorithms can we use to numerically solve these problems? When and how can we prove that the solutions returned by the algorithms are correct? These questions are relatively well understood for convex inverse problems, but the course will be on non-convex inverse problems, whose study is much more recent, and a very active research topic.

   

  The course will be at the interface between real analysis, statistics and optimization. It will include theoretical and programming exercises.

  Learning outcomes :

  Understand what is a non-convex inverse problems; get some familiarity with the most classical algorithms to solve them.

• Mathematics of deep learning

  Mathematics of deep learning

  Ects : 4

  Total hours : 24
• Topological Data Analysis

  Topological Data Analysis

  Ects : 4

  Total hours : 18
• Deep learning for image analysis

  Deep learning for image analysis

  Ects : 4

  Lecturer :
  Etienne DECENCIERE
  

  Total hours : 24
• Dimension reduction and manifold learning

  Dimension reduction and manifold learning

  Ects : 4

  Total hours : 24

  Overview :

  Modern machine learning typically deals with high-dimensional data. The fields concerned are very varied and include genomics, image, text, time series, or even socioeconomic data where more and more unstructured features are routinely collected. As a counterpart of this tendency towards exhaustiveness, understanding these data raises challenges in terms of computational resources and human understandability. Manifold Learning refers to a family of methods aiming at reducing the dimension of data while preserving certain of its geometric and structural characteristics. It is widely used in machine learning and experimental science to compress, visualize and interpret high-dimensional data. This course will provide a global overview of the methodology of the field, while focusing on the mathematical aspects underlying the techniques used in practice.

  Require prerequisites :

  Linear algebra, basic probability theory, statistics, Python coding

  Learning outcomes :

  - Curse of dimensionality, manifold hypothesis and intrinsic dimension(s) - Multidimensional scaling - Linear dimension reduction (random projections, principal component analysis) - Non-linear spectral methods (kernel PCA, ISOMAP, MVU, Laplacian eigenmaps) - Ad-hoc distance-preserving methods (diffusion maps, LLE) - Probabilistic dimension reduction and clustering (SNE, UMAP) - Neural network-based dimensionality reduction

  Bibliography-recommended reading

  - Ghojogh, B., M. Crowley, F. Karray, and A. Ghodsi (2023). Elements of dimensionality reduction and manifold learning - Lee, J. A., M. Verleysen, et al. (2007). Nonlinear dimensionality reduction

• Bayesian asymptotics

  Bayesian asymptotics

  Ects : 4

  Total hours : 18

Mémoire de recherche

Bloc mémoire

• Mémoire de recherche

  Mémoire de recherche

  Ects : 20