Prè-requis
first year level knowledge in statistics and probability
Objectif du cours
Stochastic 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.
The course is naturally divided into two parts. The first part will cover the fundamentals of Itô 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.
The course has three main objectives.
The first is to present the foundational concepts of stochastic calculus and Itô 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.
On successful completion of this course, a student will be able to:
• Apply the fundamental concepts of stochastic calculus and explain their relevance in modern statistics, machine learning, and applied contexts.
• Demonstrate familiarity with diffusion processes and their applications in machine learning.
• Use stochastic calculus to understand and design modern generative models.
• Apply stochastic calculus to design and analyze contemporary sampling methods.
Organisation des séances
Lecture 1
• Review of fundamental concepts in stochastic processes.
• Introduction to standard Brownian motion.
• Properties and path regularity of Brownian motion.
Lecture 2
• Introduction to continuous-time martingales.
• Quadratic variation and Doob-Meyer decomposition.
• Martingale convergence theorems.
Lecture 3
• Stochastic integrals with respect to continuous martingales.
• Itô isometry and Itô integral properties.
Lecture 4
• Itô’s formula and applications.
• Solutions to stochastic differential equations (SDEs).
• Connections between SDEs and PDEs (Fokker–Planck, Kolmogorov).
Lecture 5
• Girsanov’s theorem and change of measure.
• Radon-Nikodym derivatives between diffusion laws.
• Applications to likelihood ratios and importance sampling.
Lecture 6
• Martingale problem formulation.
• Equivalence between martingale problems and SDEs.
• Weak vs. strong solutions.
Lecture 7
• Introduction to diffusion models in generative modeling.
• Continuous-time perspective on diffusion models.
• Sampling guarantees and theoretical underpinnings.
Lecture 8
• Diffusion-based MCMC methods.
• Unadjusted Langevin Algorithm (ULA): analysis and limitations.
• Overdamped Langevin dynamics and extensions.
Mode de validation
Oral exam and project
Références
• D. Revuz and M. Yor. Continuous martingales and Brownian motion. Third. Vol. 293. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999, pp. xiv+602. isbn: 3-540-64325-7
• Giovanni Conforti, Alain Durmus, and Marta Gentiloni Silveri. “KL convergence guarantees for score di!usion models under minimal data assumptions”. In: SIAM Journal on Mathematics of Data Science 7.1 (2025), pp. 86–109
• A. Durmus and É. Moulines. “High-dimensional Bayesian inference via the unadjusted Langevin algorithm”. In: Bernoulli 25.4A (2019), pp. 2854–2882
Thèmes abordés
This course naturally complements other courses that cover Monte Carlo methods and generative models, such as:
• Introduction to Probabilistic Graphical Models and Deep Generative Models;
• Computational Statistics;
• Bayesian Machine Learning;
• Modèles génératifs pour l’image.
In contrast to these courses, our focus will be on the theoretical foundations of a selection of widely used and highly e »cient MCMC algorithms and generative models.
Moreover, stochastic calculus plays a central role in many models in physics and biology.
This course will equip students with the essential tools to analyze such models and develop meaningful solutions.
Alain OLIVIERO-DURMUS
École Polytechnique
