Abstract

Concentration inequalities for functions of independent random variables is an area of probability theory that has witnessed a great revolution in the last few decades, and has applications in a wide variety of areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry. Roughly speaking, if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value. This course offers a host of inequalities to illustrate this rich theory.

It describes the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented.

Pre-requisites

We shall assume notions of probability theory.

1. Boucheron, S., Lugosi, G., & Massart, P. (2013). Concentration inequalities: A nonasymptotic theory of independence. OUP Oxford.
2. Chatterjee, S. (2014). Superconcentration and related topics. Springer.
3. Garling, D. J. (2007). Inequalities: a journey into linear analysis. Cambridge University Press.
4. Ledoux, M. (2005).The concentration of measure phenomenon (No. 89). American Mathematical Soc.
5. Raginsky, M., Sason, I. (2014). Concentration of Measure Inequalities in Information Theory, Communications, and Coding. Now Publishers Inc.
6. Tropp, Joel A. An introduction to matrix concentration inequalities. arXiv preprint arXiv:1501.01571 (2015).

Lecture notes http://stephane-v-boucheron.fr/part-iii-concentration/