[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-959635","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n  \u003Cdiv class=\u0022row\u0022\u003E\n    \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n      \u003Ch2 class=\u0022h1 text-left\u0022\u003E  Stochastic calculus in machine learning\n\u003C\/h2\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n          5\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemester:\n          Annualis\u00e9\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ELecture:\u003C\/span\u003E\n                                      \u003Cspan\u003E24\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                                                                        \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EAnglais\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003Enon\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n    \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EPr\u00e9requis\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003Efirst year level knowledge in statistics and probability\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003ELecture 1\u003Cbr\u003E\n\u2022 Review of fundamental concepts in stochastic processes.\u003Cbr\u003E\n\u2022 Introduction to standard Brownian motion.\u003Cbr\u003E\n\u2022 Properties and path regularity of Brownian motion.\u003C\/p\u003E\n\u003Cp\u003ELecture 2\u003Cbr\u003E\n\u2022 Introduction to continuous-time martingales.\u003Cbr\u003E\n\u2022 Quadratic variation and Doob-Meyer decomposition.\u003Cbr\u003E\n\u2022 Martingale convergence theorems.\u003C\/p\u003E\n\u003Cp\u003ELecture 3\u003Cbr\u003E\n\u2022 Stochastic integrals with respect to continuous martingales.\u003Cbr\u003E\n\u2022 It\u00f4 isometry and It\u00f4 integral properties.\u003C\/p\u003E\n\u003Cp\u003ELecture 4\u003Cbr\u003E\n\u2022 It\u00f4\u2019s formula and applications.\u003Cbr\u003E\n\u2022 Solutions to stochastic differential equations (SDEs).\u003Cbr\u003E\n\u2022 Connections between SDEs and PDEs (Fokker\u2013Planck, Kolmogorov).\u003C\/p\u003E\n\u003Cp\u003ELecture 5\u003Cbr\u003E\n\u2022 Girsanov\u2019s theorem and change of measure.\u003Cbr\u003E\n\u2022 Radon-Nikodym derivatives between diffusion laws.\u003Cbr\u003E\n\u2022 Applications to likelihood ratios and importance sampling.\u003C\/p\u003E\n\u003Cp\u003ELecture 6\u003Cbr\u003E\n\u2022 Martingale problem formulation.\u003Cbr\u003E\n\u2022 Equivalence between martingale problems and SDEs.\u003Cbr\u003E\n\u2022 Weak vs. strong solutions.\u003C\/p\u003E\n\u003Cp\u003ELecture 7\u003Cbr\u003E\n\u2022 Introduction to diffusion models in generative modeling.\u003Cbr\u003E\n\u2022 Continuous-time perspective on diffusion models.\u003Cbr\u003E\n\u2022 Sampling guarantees and theoretical underpinnings.\u003C\/p\u003E\n\u003Cp\u003ELecture 8\u003Cbr\u003E\n\u2022 Diffusion-based MCMC methods.\u003Cbr\u003E\n\u2022 Unadjusted Langevin Algorithm (ULA): analysis and limitations.\u003Cbr\u003E\n\u2022 Overdamped Langevin dynamics and extensions.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EObjectifs d\u0026#039;apprentissage\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EStochastic calculus has become an essential tool for understanding and justifying modern machine learning techniques, such as the Unadjusted Langevin Algorithm and diffusion models. To equip students with the necessary background, this course offers an introduction to stochastic calculus and its applications in machine learning, with a particular focus on sampling and generative modeling.\u003Cbr\u003E\nThe course is naturally divided into two parts. The first part will cover the fundamentals of It\u00f4 calculus, including diffusion processes and their connections to partial differential equations. The second part will introduce applications of stochastic calculus in the analysis of diffusion-based Markov Chain Monte Carlo (MCMC) methods and generative models.\u003C\/p\u003E\n\u003Cp\u003EThe course has three main objectives.\u003Cbr\u003E\nThe first is to present the foundational concepts of stochastic calculus and It\u00f4 calculus. The second is to develop familiarity with stochastic differential equations and diffusion processes. Finally, students should be able to apply these tools to the design and analysis of modern machine learning methods, such as diffusion-based MCMC and generative models.\u003Cbr\u003E\nOn successful completion of this course, a student will be able to:\u003Cbr\u003E\n\u2022 Apply the fundamental concepts of stochastic calculus and explain their relevance in modern statistics, machine learning, and applied contexts.\u003Cbr\u003E\n\u2022 Demonstrate familiarity with diffusion processes and their applications in machine learning.\u003Cbr\u003E\n\u2022 Use stochastic calculus to understand and design modern generative models.\u003Cbr\u003E\n\u2022 Apply stochastic calculus to design and analyze contemporary sampling methods.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n                                        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003ED\u00e9tail renseign\u00e9 au niveau \u00c9l\u00e9ment Constitutif\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n  \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]