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M2 course, ENS Lyon: Concentration of measure in probability and high-dimensional statistical learning


Course of the Master of Advanced Mathematics, Probability & Statistics Track.



Guillaume Aubrun, Aurélien Garivier, Rémi Gribonval

Course description

This course will introduce the notion of concentration of measure and highlight its applications, notably in high dimensional data processing and machine learning. The course will start from deviations inequalities for averages of independent variables, and illustrate their interest for the analysis of random graphs and random projections for dimension reduction. It will then be shown how other high-dimensional random functions concentrate, and what guarantees this concentration yields for randomized algorithms and machine learning procedures to learn from large training collections.


Basic knowledge of probability theory, linear algebra and analysis over the reals.


Homework, in-class exercices and final exam: 50%. Presentation of a research article: 50% (details to come, see official page)


  1. Concentration Inequalities, by Stéphane Boucheron, Pascal Massart and Gabor Lugosi
  2. High-Dimensional Probability – An Introduction with Applications in Data Science, by Roman Vershynin
  3. Understanding Machine Learning, From Theory to Algorithms, by Shai Shalev-Shwartz and Shai Ben-David
  4. Foundations of Machine Learning by Mehryar Mohri, Afshin Rostamizadeh, and Ameet Talwalkar